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Vincent Selhorst-Jones

Vincent Selhorst-Jones

Vectors

Slide Duration:

Table of Contents

I. Introduction
Introduction to Precalculus

10m 3s

Intro
0:00
Title of the Course
0:06
Different Names for the Course
0:07
Precalculus
0:12
Math Analysis
0:14
Trigonometry
0:16
Algebra III
0:20
Geometry II
0:24
College Algebra
0:30
Same Concepts
0:36
How do the Lessons Work?
0:54
Introducing Concepts
0:56
Apply Concepts
1:04
Go through Examples
1:25
Who is this Course For?
1:38
Those Who Need eExtra Help with Class Work
1:52
Those Working on Material but not in Formal Class at School
1:54
Those Who Want a Refresher
2:00
Try to Watch the Whole Lesson
2:20
Understanding is So Important
3:56
What to Watch First
5:26
Lesson #2: Sets, Elements, and Numbers
5:30
Lesson #7: Idea of a Function
5:33
Lesson #6: Word Problems
6:04
What to Watch First, cont.
6:46
Lesson #2: Sets, Elements and Numbers
6:56
Lesson #3: Variables, Equations, and Algebra
6:58
Lesson #4: Coordinate Systems
7:00
Lesson #5: Midpoint, Distance, the Pythagorean Theorem and Slope
7:02
Lesson #6: Word Problems
7:10
Lesson #7: Idea of a Function
7:12
Lesson #8: Graphs
7:14
Graphing Calculator Appendix
7:40
What to Watch Last
8:46
Let's get Started!
9:48
Sets, Elements, & Numbers

45m 11s

Intro
0:00
Introduction
0:05
Sets and Elements
1:19
Set
1:20
Element
1:23
Name a Set
2:20
Order The Elements Appear In Has No Effect on the Set
2:55
Describing/ Defining Sets
3:28
Directly Say All the Elements
3:36
Clearly Describing All the Members of the Set
3:55
Describing the Quality (or Qualities) Each member Of the Set Has In Common
4:32
Symbols: 'Element of' and 'Subset of'
6:01
Symbol is ∈
6:03
Subset Symbol is ⊂
6:35
Empty Set
8:07
Symbol is ∅
8:20
Since It's Empty, It is a Subset of All Sets
8:44
Union and Intersection
9:54
Union Symbol is ∪
10:08
Intersection Symbol is ∩
10:18
Sets Can Be Weird Stuff
12:26
Can Have Elements in a Set
12:50
We Can Have Infinite Sets
13:09
Example
13:22
Consider a Set Where We Take a Word and Then Repeat It An Ever Increasing Number of Times
14:08
This Set Has Infinitely Many Distinct Elements
14:40
Numbers as Sets
16:03
Natural Numbers ℕ
16:16
Including 0 and the Negatives ℤ
18:13
Rational Numbers ℚ
19:27
Can Express Rational Numbers with Decimal Expansions
22:05
Irrational Numbers
23:37
Real Numbers ℝ: Put the Rational and Irrational Numbers Together
25:15
Interval Notation and the Real Numbers
26:45
Include the End Numbers
27:06
Exclude the End Numbers
27:33
Example
28:28
Interval Notation: Infinity
29:09
Use -∞ or ∞ to Show an Interval Going on Forever in One Direction or the Other
29:14
Always Use Parentheses
29:50
Examples
30:27
Example 1
31:23
Example 2
35:26
Example 3
38:02
Example 4
42:21
Variables, Equations, & Algebra

35m 31s

Intro
0:00
What is a Variable?
0:05
A Variable is a Placeholder for a Number
0:11
Affects the Output of a Function or a Dependent Variable
0:24
Naming Variables
1:51
Useful to Use Symbols
2:21
What is a Constant?
4:14
A Constant is a Fixed, Unchanging Number
4:28
We Might Refer to a Symbol Representing a Number as a Constant
4:51
What is a Coefficient?
5:33
A Coefficient is a Multiplicative Factor on a Variable
5:37
Not All Coefficients are Constants
5:51
Expressions and Equations
6:42
An Expression is a String of Mathematical Symbols That Make Sense Used Together
7:05
An Equation is a Statement That Two Expression Have the Same Value
8:20
The Idea of Algebra
8:51
Equality
8:59
If Two Things Are the Same *Equal), Then We Can Do the Exact Same Operation to Both and the Results Will Be the Same
9:41
Always Do The Exact Same Thing to Both Sides
12:22
Solving Equations
13:23
When You Are Asked to Solve an Equation, You Are Being Asked to Solve for Something
13:33
Look For What Values Makes the Equation True
13:38
Isolate the Variable by Doing Algebra
14:37
Order of Operations
16:02
Why Certain Operations are Grouped
17:01
When You Don't Have to Worry About Order
17:39
Distributive Property
18:15
It Allows Multiplication to Act Over Addition in Parentheses
18:23
We Can Use the Distributive Property in Reverse to Combine Like Terms
19:05
Substitution
20:03
Use Information From One Equation in Another Equation
20:07
Put Your Substitution in Parentheses
20:44
Example 1
23:17
Example 2
25:49
Example 3
28:11
Example 4
30:02
Coordinate Systems

35m 2s

Intro
0:00
Inherent Order in ℝ
0:05
Real Numbers Come with an Inherent Order
0:11
Positive Numbers
0:21
Negative Numbers
0:58
'Less Than' and 'Greater Than'
2:04
Tip To Help You Remember the Signs
2:56
Inequality
4:06
Less Than or Equal and Greater Than or Equal
4:51
One Dimension: The Number Line
5:36
Graphically Represent ℝ on a Number Line
5:43
Note on Infinities
5:57
With the Number Line, We Can Directly See the Order We Put on ℝ
6:35
Ordered Pairs
7:22
Example
7:34
Allows Us to Talk About Two Numbers at the Same Time
9:41
Ordered Pairs of Real Numbers Cannot be Put Into an Order Like we Did with ℝ
10:41
Two Dimensions: The Plane
13:13
We Can Represent Ordered Pairs with the Plane
13:24
Intersection is known as the Origin
14:31
Plotting the Point
14:32
Plane = Coordinate Plane = Cartesian Plane = ℝ²
17:46
The Plane and Quadrants
18:50
Quadrant I
19:04
Quadrant II
19:21
Quadrant III
20:04
Quadrant IV
20:20
Three Dimensions: Space
21:02
Create Ordered Triplets
21:09
Visually Represent This
21:19
Three-Dimension = Space = ℝ³
21:47
Higher Dimensions
22:24
If We Have n Dimensions, We Call It n-Dimensional Space or ℝ to the nth Power
22:31
We Can Represent Places In This n-Dimensional Space As Ordered Groupings of n Numbers
22:41
Hard to Visualize Higher Dimensional Spaces
23:18
Example 1
25:07
Example 2
26:10
Example 3
28:58
Example 4
31:05
Midpoints, Distance, the Pythagorean Theorem, & Slope

48m 43s

Intro
0:00
Introduction
0:07
Midpoint: One Dimension
2:09
Example of Something More Complex
2:31
Use the Idea of a Middle
3:28
Find the Midpoint of Arbitrary Values a and b
4:17
How They're Equivalent
5:05
Official Midpoint Formula
5:46
Midpoint: Two Dimensions
6:19
The Midpoint Must Occur at the Horizontal Middle and the Vertical Middle
6:38
Arbitrary Pair of Points Example
7:25
Distance: One Dimension
9:26
Absolute Value
10:54
Idea of Forcing Positive
11:06
Distance: One Dimension, Formula
11:47
Distance Between Arbitrary a and b
11:48
Absolute Value Helps When the Distance is Negative
12:41
Distance Formula
12:58
The Pythagorean Theorem
13:24
a²+b²=c²
13:50
Distance: Two Dimensions
14:59
Break Into Horizontal and Vertical Parts and then Use the Pythagorean Theorem
15:16
Distance Between Arbitrary Points (x₁,y₁) and (x₂,y₂)
16:21
Slope
19:30
Slope is the Rate of Change
19:41
m = rise over run
21:27
Slope Between Arbitrary Points (x₁,y₁) and (x₂,y₂)
22:31
Interpreting Slope
24:12
Positive Slope and Negative Slope
25:40
m=1, m=0, m=-1
26:48
Example 1
28:25
Example 2
31:42
Example 3
36:40
Example 4
42:48
Word Problems

56m 31s

Intro
0:00
Introduction
0:05
What is a Word Problem?
0:45
Describes Any Problem That Primarily Gets Its Ideas Across With Words Instead of Math Symbols
0:48
Requires Us to Think
1:32
Why Are They So Hard?
2:11
Reason 1: No Simple Formula to Solve Them
2:16
Reason 2: Harder to Teach Word Problems
2:47
You Can Learn How to Do Them!
3:51
Grades
7:57
'But I'm Never Going to Use This In Real Life'
9:46
Solving Word Problems
12:58
First: Understand the Problem
13:37
Second: What Are You Looking For?
14:33
Third: Set Up Relationships
16:21
Fourth: Solve It!
17:48
Summary of Method
19:04
Examples on Things Other Than Math
20:21
Math-Specific Method: What You Need Now
25:30
Understand What the Problem is Talking About
25:37
Set Up and Name Any Variables You Need to Know
25:56
Set Up Equations Connecting Those Variables to the Information in the Problem Statement
26:02
Use the Equations to Solve for an Answer
26:14
Tip
26:58
Draw Pictures
27:22
Breaking Into Pieces
28:28
Try Out Hypothetical Numbers
29:52
Student Logic
31:27
Jump In!
32:40
Example 1
34:03
Example 2
39:15
Example 3
44:22
Example 4
50:24
II. Functions
Idea of a Function

39m 54s

Intro
0:00
Introduction
0:04
What is a Function?
1:06
A Visual Example and Non-Example
1:30
Function Notation
3:47
f(x)
4:05
Express What Sets the Function Acts On
5:45
Metaphors for a Function
6:17
Transformation
6:28
Map
7:17
Machine
8:56
Same Input Always Gives Same Output
10:01
If We Put the Same Input Into a Function, It Will Always Produce the Same Output
10:11
Example of Something That is Not a Function
11:10
A Non-Numerical Example
12:10
The Functions We Will Use
15:05
Unless Told Otherwise, We Will Assume Every Function Takes in Real Numbers and Outputs Real Numbers
15:11
Usually Told the Rule of a Given Function
15:27
How To Use a Function
16:18
Apply the Rule to Whatever Our Input Value Is
16:28
Make Sure to Wrap Your Substitutions in Parentheses
17:09
Functions and Tables
17:36
Table of Values, Sometimes Called a T-Table
17:46
Example
17:56
Domain: What Goes In
18:55
The Domain is the Set of all Inputs That the Function Can Accept
18:56
Example
19:40
Range: What Comes Out
21:27
The Range is the Set of All Possible Outputs a Function Can Assign
21:34
Example
21:49
Another Example Would Be Our Initial Function From Earlier in This Lesson
22:29
Example 1
23:45
Example 2
25:22
Example 3
27:27
Example 4
29:23
Example 5
33:33
Graphs

58m 26s

Intro
0:00
Introduction
0:04
How to Interpret Graphs
1:17
Input / Independent Variable
1:47
Output / Dependent Variable
2:00
Graph as Input ⇒ Output
2:23
One Way to Think of a Graph: See What Happened to Various Inputs
2:25
Example
2:47
Graph as Location of Solution
4:20
A Way to See Solutions
4:36
Example
5:20
Which Way Should We Interpret?
7:13
Easiest to Think In Terms of How Inputs Are Mapped to Outputs
7:20
Sometimes It's Easier to Think In Terms of Solutions
8:39
Pay Attention to Axes
9:50
Axes Tell Where the Graph Is and What Scale It Has
10:09
Often, The Axes Will Be Square
10:14
Example
12:06
Arrows or No Arrows?
16:07
Will Not Use Arrows at the End of Our Graphs
17:13
Graph Stops Because It Hits the Edge of the Graphing Axes, Not Because the Function Stops
17:18
How to Graph
19:47
Plot Points
20:07
Connect with Curves
21:09
If You Connect with Straight Lines
21:44
Graphs of Functions are Smooth
22:21
More Points ⇒ More Accurate
23:38
Vertical Line Test
27:44
If a Vertical Line Could Intersect More Than One Point On a Graph, It Can Not Be the Graph of a Function
28:41
Every Point on a Graph Tells Us Where the x-Value Below is Mapped
30:07
Domain in Graphs
31:37
The Domain is the Set of All Inputs That a Function Can Accept
31:44
Be Aware That Our Function Probably Continues Past the Edge of Our 'Viewing Window'
33:19
Range in Graphs
33:53
Graphing Calculators: Check the Appendix!
36:55
Example 1
38:37
Example 2
45:19
Example 3
50:41
Example 4
53:28
Example 5
55:50
Properties of Functions

48m 49s

Intro
0:00
Introduction
0:05
Increasing Decreasing Constant
0:43
Looking at a Specific Graph
1:15
Increasing Interval
2:39
Constant Function
4:15
Decreasing Interval
5:10
Find Intervals by Looking at the Graph
5:32
Intervals Show x-values; Write in Parentheses
6:39
Maximum and Minimums
8:48
Relative (Local) Max/Min
10:20
Formal Definition of Relative Maximum
12:44
Formal Definition of Relative Minimum
13:05
Max/Min, More Terms
14:18
Definition of Extrema
15:01
Average Rate of Change
16:11
Drawing a Line for the Average Rate
16:48
Using the Slope of the Secant Line
17:36
Slope in Function Notation
18:45
Zeros/Roots/x-intercepts
19:45
What Zeros in a Function Mean
20:25
Even Functions
22:30
Odd Functions
24:36
Even/Odd Functions and Graphs
26:28
Example of an Even Function
27:12
Example of an Odd Function
28:03
Example 1
29:35
Example 2
33:07
Example 3
40:32
Example 4
42:34
Function Petting Zoo

29m 20s

Intro
0:00
Introduction
0:04
Don't Forget that Axes Matter!
1:44
The Constant Function
2:40
The Identity Function
3:44
The Square Function
4:40
The Cube Function
5:44
The Square Root Function
6:51
The Reciprocal Function
8:11
The Absolute Value Function
10:19
The Trigonometric Functions
11:56
f(x)=sin(x)
12:12
f(x)=cos(x)
12:24
Alternate Axes
12:40
The Exponential and Logarithmic Functions
13:35
Exponential Functions
13:44
Logarithmic Functions
14:24
Alternating Axes
15:17
Transformations and Compositions
16:08
Example 1
17:52
Example 2
18:33
Example 3
20:24
Example 4
26:07
Transformation of Functions

48m 35s

Intro
0:00
Introduction
0:04
Vertical Shift
1:12
Graphical Example
1:21
A Further Explanation
2:16
Vertical Stretch/Shrink
3:34
Graph Shrinks
3:46
Graph Stretches
3:51
A Further Explanation
5:07
Horizontal Shift
6:49
Moving the Graph to the Right
7:28
Moving the Graph to the Left
8:12
A Further Explanation
8:19
Understanding Movement on the x-axis
8:38
Horizontal Stretch/Shrink
12:59
Shrinking the Graph
13:40
Stretching the Graph
13:48
A Further Explanation
13:55
Understanding Stretches from the x-axis
14:12
Vertical Flip (aka Mirror)
16:55
Example Graph
17:07
Multiplying the Vertical Component by -1
17:18
Horizontal Flip (aka Mirror)
18:43
Example Graph
19:01
Multiplying the Horizontal Component by -1
19:54
Summary of Transformations
22:11
Stacking Transformations
24:46
Order Matters
25:20
Transformation Example
25:52
Example 1
29:21
Example 2
34:44
Example 3
38:10
Example 4
43:46
Composite Functions

33m 24s

Intro
0:00
Introduction
0:04
Arithmetic Combinations
0:40
Basic Operations
1:20
Definition of the Four Arithmetic Combinations
1:40
Composite Functions
2:53
The Function as a Machine
3:32
Function Compositions as Multiple Machines
3:59
Notation for Composite Functions
4:46
Two Formats
6:02
Another Visual Interpretation
7:17
How to Use Composite Functions
8:21
Example of on Function acting on Another
9:17
Example 1
11:03
Example 2
15:27
Example 3
21:11
Example 4
27:06
Piecewise Functions

51m 42s

Intro
0:00
Introduction
0:04
Analogies to a Piecewise Function
1:16
Different Potatoes
1:41
Factory Production
2:27
Notations for Piecewise Functions
3:39
Notation Examples from Analogies
6:11
Example of a Piecewise (with Table)
7:24
Example of a Non-Numerical Piecewise
11:35
Graphing Piecewise Functions
14:15
Graphing Piecewise Functions, Example
16:26
Continuous Functions
16:57
Statements of Continuity
19:30
Example of Continuous and Non-Continuous Graphs
20:05
Interesting Functions: the Step Function
22:00
Notation for the Step Function
22:40
How the Step Function Works
22:56
Graph of the Step Function
25:30
Example 1
26:22
Example 2
28:49
Example 3
36:50
Example 4
46:11
Inverse Functions

49m 37s

Intro
0:00
Introduction
0:04
Analogy by picture
1:10
How to Denote the inverse
1:40
What Comes out of the Inverse
1:52
Requirement for Reversing
2:02
The Basketball Factory
2:12
The Importance of Information
2:45
One-to-One
4:04
Requirement for Reversibility
4:21
When a Function has an Inverse
4:43
One-to-One
5:13
Not One-to-One
5:50
Not a Function
6:19
Horizontal Line Test
7:01
How to the test Works
7:12
One-to-One
8:12
Not One-to-One
8:45
Definition: Inverse Function
9:12
Formal Definition
9:21
Caution to Students
10:02
Domain and Range
11:12
Finding the Range of the Function Inverse
11:56
Finding the Domain of the Function Inverse
12:11
Inverse of an Inverse
13:09
Its just x!
13:26
Proof
14:03
Graphical Interpretation
17:07
Horizontal Line Test
17:20
Graph of the Inverse
18:04
Swapping Inputs and Outputs to Draw Inverses
19:02
How to Find the Inverse
21:03
What We Are Looking For
21:21
Reversing the Function
21:38
A Method to Find Inverses
22:33
Check Function is One-to-One
23:04
Swap f(x) for y
23:25
Interchange x and y
23:41
Solve for y
24:12
Replace y with the inverse
24:40
Some Comments
25:01
Keeping Step 2 and 3 Straight
25:44
Switching to Inverse
26:12
Checking Inverses
28:52
How to Check an Inverse
29:06
Quick Example of How to Check
29:56
Example 1
31:48
Example 2
34:56
Example 3
39:29
Example 4
46:19
Variation Direct and Inverse

28m 49s

Intro
0:00
Introduction
0:06
Direct Variation
1:14
Same Direction
1:21
Common Example: Groceries
1:56
Different Ways to Say that Two Things Vary Directly
2:28
Basic Equation for Direct Variation
2:55
Inverse Variation
3:40
Opposite Direction
3:50
Common Example: Gravity
4:53
Different Ways to Say that Two Things Vary Indirectly
5:48
Basic Equation for Indirect Variation
6:33
Joint Variation
7:27
Equation for Joint Variation
7:53
Explanation of the Constant
8:48
Combined Variation
9:35
Gas Law as a Combination
9:44
Single Constant
10:33
Example 1
10:49
Example 2
13:34
Example 3
15:39
Example 4
19:48
III. Polynomials
Intro to Polynomials

38m 41s

Intro
0:00
Introduction
0:04
Definition of a Polynomial
1:04
Starting Integer
2:06
Structure of a Polynomial
2:49
The a Constants
3:34
Polynomial Function
5:13
Polynomial Equation
5:23
Polynomials with Different Variables
5:36
Degree
6:23
Informal Definition
6:31
Find the Largest Exponent Variable
6:44
Quick Examples
7:36
Special Names for Polynomials
8:59
Based on the Degree
9:23
Based on the Number of Terms
10:12
Distributive Property (aka 'FOIL')
11:37
Basic Distributive Property
12:21
Distributing Two Binomials
12:55
Longer Parentheses
15:12
Reverse: Factoring
17:26
Long-Term Behavior of Polynomials
17:48
Examples
18:13
Controlling Term--Term with the Largest Exponent
19:33
Positive and Negative Coefficients on the Controlling Term
20:21
Leading Coefficient Test
22:07
Even Degree, Positive Coefficient
22:13
Even Degree, Negative Coefficient
22:39
Odd Degree, Positive Coefficient
23:09
Odd Degree, Negative Coefficient
23:27
Example 1
25:11
Example 2
27:16
Example 3
31:16
Example 4
34:41
Roots (Zeros) of Polynomials

41m 7s

Intro
0:00
Introduction
0:05
Roots in Graphs
1:17
The x-intercepts
1:33
How to Remember What 'Roots' Are
1:50
Naïve Attempts
2:31
Isolating Variables
2:45
Failures of Isolating Variables
3:30
Missing Solutions
4:59
Factoring: How to Find Roots
6:28
How Factoring Works
6:36
Why Factoring Works
7:20
Steps to Finding Polynomial Roots
9:21
Factoring: How to Find Roots CAUTION
10:08
Factoring is Not Easy
11:32
Factoring Quadratics
13:08
Quadratic Trinomials
13:21
Form of Factored Binomials
13:38
Factoring Examples
14:40
Factoring Quadratics, Check Your Work
16:58
Factoring Higher Degree Polynomials
18:19
Factoring a Cubic
18:32
Factoring a Quadratic
19:04
Factoring: Roots Imply Factors
19:54
Where a Root is, A Factor Is
20:01
How to Use Known Roots to Make Factoring Easier
20:35
Not all Polynomials Can be Factored
22:30
Irreducible Polynomials
23:27
Complex Numbers Help
23:55
Max Number of Roots/Factors
24:57
Limit to Number of Roots Equal to the Degree
25:18
Why there is a Limit
25:25
Max Number of Peaks/Valleys
26:39
Shape Information from Degree
26:46
Example Graph
26:54
Max, But Not Required
28:00
Example 1
28:37
Example 2
31:21
Example 3
36:12
Example 4
38:40
Completing the Square and the Quadratic Formula

39m 43s

Intro
0:00
Introduction
0:05
Square Roots and Equations
0:51
Taking the Square Root to Find the Value of x
0:55
Getting the Positive and Negative Answers
1:05
Completing the Square: Motivation
2:04
Polynomials that are Easy to Solve
2:20
Making Complex Polynomials Easy to Solve
3:03
Steps to Completing the Square
4:30
Completing the Square: Method
7:22
Move C over
7:35
Divide by A
7:44
Find r
7:59
Add to Both Sides to Complete the Square
8:49
Solving Quadratics with Ease
9:56
The Quadratic Formula
11:38
Derivation
11:43
Final Form
12:23
Follow Format to Use Formula
13:38
How Many Roots?
14:53
The Discriminant
15:47
What the Discriminant Tells Us: How Many Roots
15:58
How the Discriminant Works
16:30
Example 1: Complete the Square
18:24
Example 2: Solve the Quadratic
22:00
Example 3: Solve for Zeroes
25:28
Example 4: Using the Quadratic Formula
30:52
Properties of Quadratic Functions

45m 34s

Intro
0:00
Introduction
0:05
Parabolas
0:35
Examples of Different Parabolas
1:06
Axis of Symmetry and Vertex
1:28
Drawing an Axis of Symmetry
1:51
Placing the Vertex
2:28
Looking at the Axis of Symmetry and Vertex for other Parabolas
3:09
Transformations
4:18
Reviewing Transformation Rules
6:28
Note the Different Horizontal Shift Form
7:45
An Alternate Form to Quadratics
8:54
The Constants: k, h, a
9:05
Transformations Formed
10:01
Analyzing Different Parabolas
10:10
Switching Forms by Completing the Square
11:43
Vertex of a Parabola
16:30
Vertex at (h, k)
16:47
Vertex in Terms of a, b, and c Coefficients
17:28
Minimum/Maximum at Vertex
18:19
When a is Positive
18:25
When a is Negative
18:52
Axis of Symmetry
19:54
Incredibly Minor Note on Grammar
20:52
Example 1
21:48
Example 2
26:35
Example 3
28:55
Example 4
31:40
Intermediate Value Theorem and Polynomial Division

46m 8s

Intro
0:00
Introduction
0:05
Reminder: Roots Imply Factors
1:32
The Intermediate Value Theorem
3:41
The Basis: U between a and b
4:11
U is on the Function
4:52
Intermediate Value Theorem, Proof Sketch
5:51
If Not True, the Graph Would Have to Jump
5:58
But Graph is Defined as Continuous
6:43
Finding Roots with the Intermediate Value Theorem
7:01
Picking a and b to be of Different Signs
7:10
Must Be at Least One Root
7:46
Dividing a Polynomial
8:16
Using Roots and Division to Factor
8:38
Long Division Refresher
9:08
The Division Algorithm
12:18
How It Works to Divide Polynomials
12:37
The Parts of the Equation
13:24
Rewriting the Equation
14:47
Polynomial Long Division
16:20
Polynomial Long Division In Action
16:29
One Step at a Time
20:51
Synthetic Division
22:46
Setup
23:11
Synthetic Division, Example
24:44
Which Method Should We Use
26:39
Advantages of Synthetic Method
26:49
Advantages of Long Division
27:13
Example 1
29:24
Example 2
31:27
Example 3
36:22
Example 4
40:55
Complex Numbers

45m 36s

Intro
0:00
Introduction
0:04
A Wacky Idea
1:02
The Definition of the Imaginary Number
1:22
How it Helps Solve Equations
2:20
Square Roots and Imaginary Numbers
3:15
Complex Numbers
5:00
Real Part and Imaginary Part
5:20
When Two Complex Numbers are Equal
6:10
Addition and Subtraction
6:40
Deal with Real and Imaginary Parts Separately
7:36
Two Quick Examples
7:54
Multiplication
9:07
FOIL Expansion
9:14
Note What Happens to the Square of the Imaginary Number
9:41
Two Quick Examples
10:22
Division
11:27
Complex Conjugates
13:37
Getting Rid of i
14:08
How to Denote the Conjugate
14:48
Division through Complex Conjugates
16:11
Multiply by the Conjugate of the Denominator
16:28
Example
17:46
Factoring So-Called 'Irreducible' Quadratics
19:24
Revisiting the Quadratic Formula
20:12
Conjugate Pairs
20:37
But Are the Complex Numbers 'Real'?
21:27
What Makes a Number Legitimate
25:38
Where Complex Numbers are Used
27:20
Still, We Won't See Much of C
29:05
Example 1
30:30
Example 2
33:15
Example 3
38:12
Example 4
42:07
Fundamental Theorem of Algebra

19m 9s

Intro
0:00
Introduction
0:05
Idea: Hidden Roots
1:16
Roots in Complex Form
1:42
All Polynomials Have Roots
2:08
Fundamental Theorem of Algebra
2:21
Where Are All the Imaginary Roots, Then?
3:17
All Roots are Complex
3:45
Real Numbers are a Subset of Complex Numbers
3:59
The n Roots Theorem
5:01
For Any Polynomial, Its Degree is Equal to the Number of Roots
5:11
Equivalent Statement
5:24
Comments: Multiplicity
6:29
Non-Distinct Roots
6:59
Denoting Multiplicity
7:20
Comments: Complex Numbers Necessary
7:41
Comments: Complex Coefficients Allowed
8:55
Comments: Existence Theorem
9:59
Proof Sketch of n Roots Theorem
10:45
First Root
11:36
Second Root
13:23
Continuation to Find all Roots
16:00
IV. Rational Functions
Rational Functions and Vertical Asymptotes

33m 22s

Intro
0:00
Introduction
0:05
Definition of a Rational Function
1:20
Examples of Rational Functions
2:30
Why They are Called 'Rational'
2:47
Domain of a Rational Function
3:15
Undefined at Denominator Zeros
3:25
Otherwise all Reals
4:16
Investigating a Fundamental Function
4:50
The Domain of the Function
5:04
What Occurs at the Zeroes of the Denominator
5:20
Idea of a Vertical Asymptote
6:23
What's Going On?
6:58
Approaching x=0 from the left
7:32
Approaching x=0 from the right
8:34
Dividing by Very Small Numbers Results in Very Large Numbers
9:31
Definition of a Vertical Asymptote
10:05
Vertical Asymptotes and Graphs
11:15
Drawing Asymptotes by Using a Dashed Line
11:27
The Graph Can Never Touch Its Undefined Point
12:00
Not All Zeros Give Asymptotes
13:02
Special Cases: When Numerator and Denominator Go to Zero at the Same Time
14:58
Cancel out Common Factors
15:49
How to Find Vertical Asymptotes
16:10
Figure out What Values Are Not in the Domain of x
16:24
Determine if the Numerator and Denominator Share Common Factors and Cancel
16:45
Find Denominator Roots
17:33
Note if Asymptote Approaches Negative or Positive Infinity
18:06
Example 1
18:57
Example 2
21:26
Example 3
23:04
Example 4
30:01
Horizontal Asymptotes

34m 16s

Intro
0:00
Introduction
0:05
Investigating a Fundamental Function
0:53
What Happens as x Grows Large
1:00
Different View
1:12
Idea of a Horizontal Asymptote
1:36
What's Going On?
2:24
What Happens as x Grows to a Large Negative Number
2:49
What Happens as x Grows to a Large Number
3:30
Dividing by Very Large Numbers Results in Very Small Numbers
3:52
Example Function
4:41
Definition of a Vertical Asymptote
8:09
Expanding the Idea
9:03
What's Going On?
9:48
What Happens to the Function in the Long Run?
9:51
Rewriting the Function
10:13
Definition of a Slant Asymptote
12:09
Symbolical Definition
12:30
Informal Definition
12:45
Beyond Slant Asymptotes
13:03
Not Going Beyond Slant Asymptotes
14:39
Horizontal/Slant Asymptotes and Graphs
15:43
How to Find Horizontal and Slant Asymptotes
16:52
How to Find Horizontal Asymptotes
17:12
Expand the Given Polynomials
17:18
Compare the Degrees of the Numerator and Denominator
17:40
How to Find Slant Asymptotes
20:05
Slant Asymptotes Exist When n+m=1
20:08
Use Polynomial Division
20:24
Example 1
24:32
Example 2
25:53
Example 3
26:55
Example 4
29:22
Graphing Asymptotes in a Nutshell

49m 7s

Intro
0:00
Introduction
0:05
A Process for Graphing
1:22
1. Factor Numerator and Denominator
1:50
2. Find Domain
2:53
3. Simplifying the Function
3:59
4. Find Vertical Asymptotes
4:59
5. Find Horizontal/Slant Asymptotes
5:24
6. Find Intercepts
7:35
7. Draw Graph (Find Points as Necessary)
9:21
Draw Graph Example
11:21
Vertical Asymptote
11:41
Horizontal Asymptote
11:50
Other Graphing
12:16
Test Intervals
15:08
Example 1
17:57
Example 2
23:01
Example 3
29:02
Example 4
33:37
Partial Fractions

44m 56s

Intro
0:00
Introduction: Idea
0:04
Introduction: Prerequisites and Uses
1:57
Proper vs. Improper Polynomial Fractions
3:11
Possible Things in the Denominator
4:38
Linear Factors
6:16
Example of Linear Factors
7:03
Multiple Linear Factors
7:48
Irreducible Quadratic Factors
8:25
Example of Quadratic Factors
9:26
Multiple Quadratic Factors
9:49
Mixing Factor Types
10:28
Figuring Out the Numerator
11:10
How to Solve for the Constants
11:30
Quick Example
11:40
Example 1
14:29
Example 2
18:35
Example 3
20:33
Example 4
28:51
V. Exponential & Logarithmic Functions
Understanding Exponents

35m 17s

Intro
0:00
Introduction
0:05
Fundamental Idea
1:46
Expanding the Idea
2:28
Multiplication of the Same Base
2:40
Exponents acting on Exponents
3:45
Different Bases with the Same Exponent
4:31
To the Zero
5:35
To the First
5:45
Fundamental Rule with the Zero Power
6:35
To the Negative
7:45
Any Number to a Negative Power
8:14
A Fraction to a Negative Power
9:58
Division with Exponential Terms
10:41
To the Fraction
11:33
Square Root
11:58
Any Root
12:59
Summary of Rules
14:38
To the Irrational
17:21
Example 1
20:34
Example 2
23:42
Example 3
27:44
Example 4
31:44
Example 5
33:15
Exponential Functions

47m 4s

Intro
0:00
Introduction
0:05
Definition of an Exponential Function
0:48
Definition of the Base
1:02
Restrictions on the Base
1:16
Computing Exponential Functions
2:29
Harder Computations
3:10
When to Use a Calculator
3:21
Graphing Exponential Functions: a>1
6:02
Three Examples
6:13
What to Notice on the Graph
7:44
A Story
8:27
Story Diagram
9:15
Increasing Exponentials
11:29
Story Morals
14:40
Application: Compound Interest
15:15
Compounding Year after Year
16:01
Function for Compounding Interest
16:51
A Special Number: e
20:55
Expression for e
21:28
Where e stabilizes
21:55
Application: Continuously Compounded Interest
24:07
Equation for Continuous Compounding
24:22
Exponential Decay 0<a<1
25:50
Three Examples
26:11
Why they 'lose' value
26:54
Example 1
27:47
Example 2
33:11
Example 3
36:34
Example 4
41:28
Introduction to Logarithms

40m 31s

Intro
0:00
Introduction
0:04
Definition of a Logarithm, Base 2
0:51
Log 2 Defined
0:55
Examples
2:28
Definition of a Logarithm, General
3:23
Examples of Logarithms
5:15
Problems with Unusual Bases
7:38
Shorthand Notation: ln and log
9:44
base e as ln
10:01
base 10 as log
10:34
Calculating Logarithms
11:01
using a calculator
11:34
issues with other bases
11:58
Graphs of Logarithms
13:21
Three Examples
13:29
Slow Growth
15:19
Logarithms as Inverse of Exponentiation
16:02
Using Base 2
16:05
General Case
17:10
Looking More Closely at Logarithm Graphs
19:16
The Domain of Logarithms
20:41
Thinking about Logs like Inverses
21:08
The Alternate
24:00
Example 1
25:59
Example 2
30:03
Example 3
32:49
Example 4
37:34
Properties of Logarithms

42m 33s

Intro
0:00
Introduction
0:04
Basic Properties
1:12
Inverse--log(exp)
1:43
A Key Idea
2:44
What We Get through Exponentiation
3:18
B Always Exists
4:50
Inverse--exp(log)
5:53
Logarithm of a Power
7:44
Logarithm of a Product
10:07
Logarithm of a Quotient
13:48
Caution! There Is No Rule for loga(M+N)
16:12
Summary of Properties
17:42
Change of Base--Motivation
20:17
No Calculator Button
20:59
A Specific Example
21:45
Simplifying
23:45
Change of Base--Formula
24:14
Example 1
25:47
Example 2
29:08
Example 3
31:14
Example 4
34:13
Solving Exponential and Logarithmic Equations

34m 10s

Intro
0:00
Introduction
0:05
One to One Property
1:09
Exponential
1:26
Logarithmic
1:44
Specific Considerations
2:02
One-to-One Property
3:30
Solving by One-to-One
4:11
Inverse Property
6:09
Solving by Inverses
7:25
Dealing with Equations
7:50
Example of Taking an Exponent or Logarithm of an Equation
9:07
A Useful Property
11:57
Bring Down Exponents
12:01
Try to Simplify
13:20
Extraneous Solutions
13:45
Example 1
16:37
Example 2
19:39
Example 3
21:37
Example 4
26:45
Example 5
29:37
Application of Exponential and Logarithmic Functions

48m 46s

Intro
0:00
Introduction
0:06
Applications of Exponential Functions
1:07
A Secret!
2:17
Natural Exponential Growth Model
3:07
Figure out r
3:34
A Secret!--Why Does It Work?
4:44
e to the r Morphs
4:57
Example
5:06
Applications of Logarithmic Functions
8:32
Examples
8:43
What Logarithms are Useful For
9:53
Example 1
11:29
Example 2
15:30
Example 3
26:22
Example 4
32:05
Example 5
39:19
VI. Trigonometric Functions
Angles

39m 5s

Intro
0:00
Degrees
0:22
Circle is 360 Degrees
0:48
Splitting a Circle
1:13
Radians
2:08
Circle is 2 Pi Radians
2:31
One Radian
2:52
Half-Circle and Right Angle
4:00
Converting Between Degrees and Radians
6:24
Formulas for Degrees and Radians
6:52
Coterminal, Complementary, Supplementary Angles
7:23
Coterminal Angles
7:30
Complementary Angles
9:40
Supplementary Angles
10:08
Example 1: Dividing a Circle
10:38
Example 2: Converting Between Degrees and Radians
11:56
Example 3: Quadrants and Coterminal Angles
14:18
Extra Example 1: Common Angle Conversions
-1
Extra Example 2: Quadrants and Coterminal Angles
-2
Sine and Cosine Functions

43m 16s

Intro
0:00
Sine and Cosine
0:15
Unit Circle
0:22
Coordinates on Unit Circle
1:03
Right Triangles
1:52
Adjacent, Opposite, Hypotenuse
2:25
Master Right Triangle Formula: SOHCAHTOA
2:48
Odd Functions, Even Functions
4:40
Example: Odd Function
4:56
Example: Even Function
7:30
Example 1: Sine and Cosine
10:27
Example 2: Graphing Sine and Cosine Functions
14:39
Example 3: Right Triangle
21:40
Example 4: Odd, Even, or Neither
26:01
Extra Example 1: Right Triangle
-1
Extra Example 2: Graphing Sine and Cosine Functions
-2
Sine and Cosine Values of Special Angles

33m 5s

Intro
0:00
45-45-90 Triangle and 30-60-90 Triangle
0:08
45-45-90 Triangle
0:21
30-60-90 Triangle
2:06
Mnemonic: All Students Take Calculus (ASTC)
5:21
Using the Unit Circle
5:59
New Angles
6:21
Other Quadrants
9:43
Mnemonic: All Students Take Calculus
10:13
Example 1: Convert, Quadrant, Sine/Cosine
13:11
Example 2: Convert, Quadrant, Sine/Cosine
16:48
Example 3: All Angles and Quadrants
20:21
Extra Example 1: Convert, Quadrant, Sine/Cosine
-1
Extra Example 2: All Angles and Quadrants
-2
Modified Sine Waves: Asin(Bx+C)+D and Acos(Bx+C)+D

52m 3s

Intro
0:00
Amplitude and Period of a Sine Wave
0:38
Sine Wave Graph
0:58
Amplitude: Distance from Middle to Peak
1:18
Peak: Distance from Peak to Peak
2:41
Phase Shift and Vertical Shift
4:13
Phase Shift: Distance Shifted Horizontally
4:16
Vertical Shift: Distance Shifted Vertically
6:48
Example 1: Amplitude/Period/Phase and Vertical Shift
8:04
Example 2: Amplitude/Period/Phase and Vertical Shift
17:39
Example 3: Find Sine Wave Given Attributes
25:23
Extra Example 1: Amplitude/Period/Phase and Vertical Shift
-1
Extra Example 2: Find Cosine Wave Given Attributes
-2
Tangent and Cotangent Functions

36m 4s

Intro
0:00
Tangent and Cotangent Definitions
0:21
Tangent Definition
0:25
Cotangent Definition
0:47
Master Formula: SOHCAHTOA
1:01
Mnemonic
1:16
Tangent and Cotangent Values
2:29
Remember Common Values of Sine and Cosine
2:46
90 Degrees Undefined
4:36
Slope and Menmonic: ASTC
5:47
Uses of Tangent
5:54
Example: Tangent of Angle is Slope
6:09
Sign of Tangent in Quadrants
7:49
Example 1: Graph Tangent and Cotangent Functions
10:42
Example 2: Tangent and Cotangent of Angles
16:09
Example 3: Odd, Even, or Neither
18:56
Extra Example 1: Tangent and Cotangent of Angles
-1
Extra Example 2: Tangent and Cotangent of Angles
-2
Secant and Cosecant Functions

27m 18s

Intro
0:00
Secant and Cosecant Definitions
0:17
Secant Definition
0:18
Cosecant Definition
0:33
Example 1: Graph Secant Function
0:48
Example 2: Values of Secant and Cosecant
6:49
Example 3: Odd, Even, or Neither
12:49
Extra Example 1: Graph of Cosecant Function
-1
Extra Example 2: Values of Secant and Cosecant
-2
Inverse Trigonometric Functions

32m 58s

Intro
0:00
Arcsine Function
0:24
Restrictions between -1 and 1
0:43
Arcsine Notation
1:26
Arccosine Function
3:07
Restrictions between -1 and 1
3:36
Cosine Notation
3:53
Arctangent Function
4:30
Between -Pi/2 and Pi/2
4:44
Tangent Notation
5:02
Example 1: Domain/Range/Graph of Arcsine
5:45
Example 2: Arcsin/Arccos/Arctan Values
10:46
Example 3: Domain/Range/Graph of Arctangent
17:14
Extra Example 1: Domain/Range/Graph of Arccosine
-1
Extra Example 2: Arcsin/Arccos/Arctan Values
-2
Computations of Inverse Trigonometric Functions

31m 8s

Intro
0:00
Inverse Trigonometric Function Domains and Ranges
0:31
Arcsine
0:41
Arccosine
1:14
Arctangent
1:41
Example 1: Arcsines of Common Values
2:44
Example 2: Odd, Even, or Neither
5:57
Example 3: Arccosines of Common Values
12:24
Extra Example 1: Arctangents of Common Values
-1
Extra Example 2: Arcsin/Arccos/Arctan Values
-2
VII. Trigonometric Identities
Pythagorean Identity

19m 11s

Intro
0:00
Pythagorean Identity
0:17
Pythagorean Triangle
0:27
Pythagorean Identity
0:45
Example 1: Use Pythagorean Theorem to Prove Pythagorean Identity
1:14
Example 2: Find Angle Given Cosine and Quadrant
4:18
Example 3: Verify Trigonometric Identity
8:00
Extra Example 1: Use Pythagorean Identity to Prove Pythagorean Theorem
-1
Extra Example 2: Find Angle Given Cosine and Quadrant
-2
Identity Tan(squared)x+1=Sec(squared)x

23m 16s

Intro
0:00
Main Formulas
0:19
Companion to Pythagorean Identity
0:27
For Cotangents and Cosecants
0:52
How to Remember
0:58
Example 1: Prove the Identity
1:40
Example 2: Given Tan Find Sec
3:42
Example 3: Prove the Identity
7:45
Extra Example 1: Prove the Identity
-1
Extra Example 2: Given Sec Find Tan
-2
Addition and Subtraction Formulas

52m 52s

Intro
0:00
Addition and Subtraction Formulas
0:09
How to Remember
0:48
Cofunction Identities
1:31
How to Remember Graphically
1:44
Where to Use Cofunction Identities
2:52
Example 1: Derive the Formula for cos(A-B)
3:08
Example 2: Use Addition and Subtraction Formulas
16:03
Example 3: Use Addition and Subtraction Formulas to Prove Identity
25:11
Extra Example 1: Use cos(A-B) and Cofunction Identities
-1
Extra Example 2: Convert to Radians and use Formulas
-2
Double Angle Formulas

29m 5s

Intro
0:00
Main Formula
0:07
How to Remember from Addition Formula
0:18
Two Other Forms
1:35
Example 1: Find Sine and Cosine of Angle using Double Angle
3:16
Example 2: Prove Trigonometric Identity using Double Angle
9:37
Example 3: Use Addition and Subtraction Formulas
12:38
Extra Example 1: Find Sine and Cosine of Angle using Double Angle
-1
Extra Example 2: Prove Trigonometric Identity using Double Angle
-2
Half-Angle Formulas

43m 55s

Intro
0:00
Main Formulas
0:09
Confusing Part
0:34
Example 1: Find Sine and Cosine of Angle using Half-Angle
0:54
Example 2: Prove Trigonometric Identity using Half-Angle
11:51
Example 3: Prove the Half-Angle Formula for Tangents
18:39
Extra Example 1: Find Sine and Cosine of Angle using Half-Angle
-1
Extra Example 2: Prove Trigonometric Identity using Half-Angle
-2
VIII. Applications of Trigonometry
Trigonometry in Right Angles

25m 43s

Intro
0:00
Master Formula for Right Angles
0:11
SOHCAHTOA
0:15
Only for Right Triangles
1:26
Example 1: Find All Angles in a Triangle
2:19
Example 2: Find Lengths of All Sides of Triangle
7:39
Example 3: Find All Angles in a Triangle
11:00
Extra Example 1: Find All Angles in a Triangle
-1
Extra Example 2: Find Lengths of All Sides of Triangle
-2
Law of Sines

56m 40s

Intro
0:00
Law of Sines Formula
0:18
SOHCAHTOA
0:27
Any Triangle
0:59
Graphical Representation
1:25
Solving Triangle Completely
2:37
When to Use Law of Sines
2:55
ASA, SAA, SSA, AAA
2:59
SAS, SSS for Law of Cosines
7:11
Example 1: How Many Triangles Satisfy Conditions, Solve Completely
8:44
Example 2: How Many Triangles Satisfy Conditions, Solve Completely
15:30
Example 3: How Many Triangles Satisfy Conditions, Solve Completely
28:32
Extra Example 1: How Many Triangles Satisfy Conditions, Solve Completely
-1
Extra Example 2: How Many Triangles Satisfy Conditions, Solve Completely
-2
Law of Cosines

49m 5s

Intro
0:00
Law of Cosines Formula
0:23
Graphical Representation
0:34
Relates Sides to Angles
1:00
Any Triangle
1:20
Generalization of Pythagorean Theorem
1:32
When to Use Law of Cosines
2:26
SAS, SSS
2:30
Heron's Formula
4:49
Semiperimeter S
5:11
Example 1: How Many Triangles Satisfy Conditions, Solve Completely
5:53
Example 2: How Many Triangles Satisfy Conditions, Solve Completely
15:19
Example 3: Find Area of a Triangle Given All Side Lengths
26:33
Extra Example 1: How Many Triangles Satisfy Conditions, Solve Completely
-1
Extra Example 2: Length of Third Side and Area of Triangle
-2
Finding the Area of a Triangle

27m 37s

Intro
0:00
Master Right Triangle Formula and Law of Cosines
0:19
SOHCAHTOA
0:27
Law of Cosines
1:23
Heron's Formula
2:22
Semiperimeter S
2:37
Example 1: Area of Triangle with Two Sides and One Angle
3:12
Example 2: Area of Triangle with Three Sides
6:11
Example 3: Area of Triangle with Three Sides, No Heron's Formula
8:50
Extra Example 1: Area of Triangle with Two Sides and One Angle
-1
Extra Example 2: Area of Triangle with Two Sides and One Angle
-2
Word Problems and Applications of Trigonometry

34m 25s

Intro
0:00
Formulas to Remember
0:11
SOHCAHTOA
0:15
Law of Sines
0:55
Law of Cosines
1:48
Heron's Formula
2:46
Example 1: Telephone Pole Height
4:01
Example 2: Bridge Length
7:48
Example 3: Area of Triangular Field
14:20
Extra Example 1: Kite Height
-1
Extra Example 2: Roads to a Town
-2
IX. Systems of Equations and Inequalities
Systems of Linear Equations

55m 40s

Intro
0:00
Introduction
0:04
Graphs as Location of 'True'
1:49
All Locations that Make the Function True
2:25
Understand the Relationship Between Solutions and the Graph
3:43
Systems as Graphs
4:07
Equations as Lines
4:20
Intersection Point
5:19
Three Possibilities for Solutions
6:17
Independent
6:24
Inconsistent
6:36
Dependent
7:06
Solving by Substitution
8:37
Solve for One Variable
9:07
Substitute into the Second Equation
9:34
Solve for Both Variables
10:12
What If a System is Inconsistent or Dependent?
11:08
No Solutions
11:25
Infinite Solutions
12:30
Solving by Elimination
13:56
Example
14:22
Determining the Number of Solutions
16:30
Why Elimination Makes Sense
17:25
Solving by Graphing Calculator
19:59
Systems with More than Two Variables
23:22
Example 1
25:49
Example 2
30:22
Example 3
34:11
Example 4
38:55
Example 5
46:01
(Non-) Example 6
53:37
Systems of Linear Inequalities

1h 13s

Intro
0:00
Introduction
0:04
Inequality Refresher-Solutions
0:46
Equation Solutions vs. Inequality Solutions
1:02
Essentially a Wide Variety of Answers
1:35
Refresher--Negative Multiplication Flips
1:43
Refresher--Negative Flips: Why?
3:19
Multiplication by a Negative
3:43
The Relationship Flips
3:55
Refresher--Stick to Basic Operations
4:34
Linear Equations in Two Variables
6:50
Graphing Linear Inequalities
8:28
Why It Includes a Whole Section
8:43
How to Show The Difference Between Strict and Not Strict Inequalities
10:08
Dashed Line--Not Solutions
11:10
Solid Line--Are Solutions
11:24
Test Points for Shading
11:42
Example of Using a Point
12:41
Drawing Shading from the Point
13:14
Graphing a System
14:53
Set of Solutions is the Overlap
15:17
Example
15:22
Solutions are Best Found Through Graphing
18:05
Linear Programming-Idea
19:52
Use a Linear Objective Function
20:15
Variables in Objective Function have Constraints
21:24
Linear Programming-Method
22:09
Rearrange Equations
22:21
Graph
22:49
Critical Solution is at the Vertex of the Overlap
23:40
Try Each Vertice
24:35
Example 1
24:58
Example 2
28:57
Example 3
33:48
Example 4
43:10
Nonlinear Systems

41m 1s

Intro
0:00
Introduction
0:06
Substitution
1:12
Example
1:22
Elimination
3:46
Example
3:56
Elimination is Less Useful for Nonlinear Systems
4:56
Graphing
5:56
Using a Graphing Calculator
6:44
Number of Solutions
8:44
Systems of Nonlinear Inequalities
10:02
Graph Each Inequality
10:06
Dashed and/or Solid
10:18
Shade Appropriately
11:14
Example 1
13:24
Example 2
15:50
Example 3
22:02
Example 4
29:06
Example 4, cont.
33:40
X. Vectors and Matrices
Vectors

1h 9m 31s

Intro
0:00
Introduction
0:10
Magnitude of the Force
0:22
Direction of the Force
0:48
Vector
0:52
Idea of a Vector
1:30
How Vectors are Denoted
2:00
Component Form
3:20
Angle Brackets and Parentheses
3:50
Magnitude/Length
4:26
Denoting the Magnitude of a Vector
5:16
Direction/Angle
7:52
Always Draw a Picture
8:50
Component Form from Magnitude & Angle
10:10
Scaling by Scalars
14:06
Unit Vectors
16:26
Combining Vectors - Algebraically
18:10
Combining Vectors - Geometrically
19:54
Resultant Vector
20:46
Alternate Component Form: i, j
21:16
The Zero Vector
23:18
Properties of Vectors
24:20
No Multiplication (Between Vectors)
28:30
Dot Product
29:40
Motion in a Medium
30:10
Fish in an Aquarium Example
31:38
More Than Two Dimensions
33:12
More Than Two Dimensions - Magnitude
34:18
Example 1
35:26
Example 2
38:10
Example 3
45:48
Example 4
50:40
Example 4, cont.
56:07
Example 5
1:01:32
Dot Product & Cross Product

35m 20s

Intro
0:00
Introduction
0:08
Dot Product - Definition
0:42
Dot Product Results in a Scalar, Not a Vector
2:10
Example in Two Dimensions
2:34
Angle and the Dot Product
2:58
The Dot Product of Two Vectors is Deeply Related to the Angle Between the Two Vectors
2:59
Proof of Dot Product Formula
4:14
Won't Directly Help Us Better Understand Vectors
4:18
Dot Product - Geometric Interpretation
4:58
We Can Interpret the Dot Product as a Measure of How Long and How Parallel Two Vectors Are
7:26
Dot Product - Perpendicular Vectors
8:24
If the Dot Product of Two Vectors is 0, We Know They are Perpendicular to Each Other
8:54
Cross Product - Definition
11:08
Cross Product Only Works in Three Dimensions
11:09
Cross Product - A Mnemonic
12:16
The Determinant of a 3 x 3 Matrix and Standard Unit Vectors
12:17
Cross Product - Geometric Interpretations
14:30
The Right-Hand Rule
15:17
Cross Product - Geometric Interpretations Cont.
17:00
Example 1
18:40
Example 2
22:50
Example 3
24:04
Example 4
26:20
Bonus Round
29:18
Proof: Dot Product Formula
29:24
Proof: Dot Product Formula, cont.
30:38
Matrices

54m 7s

Intro
0:00
Introduction
0:08
Definition of a Matrix
3:02
Size or Dimension
3:58
Square Matrix
4:42
Denoted by Capital Letters
4:56
When are Two Matrices Equal?
5:04
Examples of Matrices
6:44
Rows x Columns
6:46
Talking About Specific Entries
7:48
We Use Capitals to Denote a Matrix and Lower Case to Denotes Its Entries
8:32
Using Entries to Talk About Matrices
10:08
Scalar Multiplication
11:26
Scalar = Real Number
11:34
Example
12:36
Matrix Addition
13:08
Example
14:22
Matrix Multiplication
15:00
Example
18:52
Matrix Multiplication, cont.
19:58
Matrix Multiplication and Order (Size)
25:26
Make Sure Their Orders are Compatible
25:27
Matrix Multiplication is NOT Commutative
28:20
Example
30:08
Special Matrices - Zero Matrix (0)
32:48
Zero Matrix Has 0 for All of its Entries
32:49
Special Matrices - Identity Matrix (I)
34:14
Identity Matrix is a Square Matrix That Has 1 for All Its Entries on the Main Diagonal and 0 for All Other Entries
34:15
Example 1
36:16
Example 2
40:00
Example 3
44:54
Example 4
50:08
Determinants & Inverses of Matrices

47m 12s

Intro
0:00
Introduction
0:06
Not All Matrices Are Invertible
1:30
What Must a Matrix Have to Be Invertible?
2:08
Determinant
2:32
The Determinant is a Real Number Associated With a Square Matrix
2:38
If the Determinant of a Matrix is Nonzero, the Matrix is Invertible
3:40
Determinant of a 2 x 2 Matrix
4:34
Think in Terms of Diagonals
5:12
Minors and Cofactors - Minors
6:24
Example
6:46
Minors and Cofactors - Cofactors
8:00
Cofactor is Closely Based on the Minor
8:01
Alternating Sign Pattern
9:04
Determinant of Larger Matrices
10:56
Example
13:00
Alternative Method for 3x3 Matrices
16:46
Not Recommended
16:48
Inverse of a 2 x 2 Matrix
19:02
Inverse of Larger Matrices
20:00
Using Inverse Matrices
21:06
When Multiplied Together, They Create the Identity Matrix
21:24
Example 1
23:45
Example 2
27:21
Example 3
32:49
Example 4
36:27
Finding the Inverse of Larger Matrices
41:59
General Inverse Method - Step 1
43:25
General Inverse Method - Step 2
43:27
General Inverse Method - Step 2, cont.
43:27
General Inverse Method - Step 3
45:15
Using Matrices to Solve Systems of Linear Equations

58m 34s

Intro
0:00
Introduction
0:12
Augmented Matrix
1:44
We Can Represent the Entire Linear System With an Augmented Matrix
1:50
Row Operations
3:22
Interchange the Locations of Two Rows
3:50
Multiply (or Divide) a Row by a Nonzero Number
3:58
Add (or Subtract) a Multiple of One Row to Another
4:12
Row Operations - Keep Notes!
5:50
Suggested Symbols
7:08
Gauss-Jordan Elimination - Idea
8:04
Gauss-Jordan Elimination - Idea, cont.
9:16
Reduced Row-Echelon Form
9:18
Gauss-Jordan Elimination - Method
11:36
Begin by Writing the System As An Augmented Matrix
11:38
Gauss-Jordan Elimination - Method, cont.
13:48
Cramer's Rule - 2 x 2 Matrices
17:08
Cramer's Rule - n x n Matrices
19:24
Solving with Inverse Matrices
21:10
Solving Inverse Matrices, cont.
25:28
The Mighty (Graphing) Calculator
26:38
Example 1
29:56
Example 2
33:56
Example 3
37:00
Example 3, cont.
45:04
Example 4
51:28
XI. Alternate Ways to Graph
Parametric Equations

53m 33s

Intro
0:00
Introduction
0:06
Definition
1:10
Plane Curve
1:24
The Key Idea
2:00
Graphing with Parametric Equations
2:52
Same Graph, Different Equations
5:04
How Is That Possible?
5:36
Same Graph, Different Equations, cont.
5:42
Here's Another to Consider
7:56
Same Plane Curve, But Still Different
8:10
A Metaphor for Parametric Equations
9:36
Think of Parametric Equations As a Way to Describe the Motion of An Object
9:38
Graph Shows Where It Went, But Not Speed
10:32
Eliminating Parameters
12:14
Rectangular Equation
12:16
Caution
13:52
Creating Parametric Equations
14:30
Interesting Graphs
16:38
Graphing Calculators, Yay!
19:18
Example 1
22:36
Example 2
28:26
Example 3
37:36
Example 4
41:00
Projectile Motion
44:26
Example 5
47:00
Polar Coordinates

48m 7s

Intro
0:00
Introduction
0:04
Polar Coordinates Give Us a Way To Describe the Location of a Point
0:26
Polar Equations and Functions
0:50
Plotting Points with Polar Coordinates
1:06
The Distance of the Point from the Origin
1:09
The Angle of the Point
1:33
Give Points as the Ordered Pair (r,θ)
2:03
Visualizing Plotting in Polar Coordinates
2:32
First Way We Can Plot
2:39
Second Way We Can Plot
2:50
First, We'll Look at Visualizing r, Then θ
3:09
Rotate the Length Counter-Clockwise by θ
3:38
Alternatively, We Can Visualize θ, Then r
4:06
'Polar Graph Paper'
6:17
Horizontal and Vertical Tick Marks Are Not Useful for Polar
6:42
Use Concentric Circles to Helps Up See Distance From the Pole
7:08
Can Use Arc Sectors to See Angles
7:57
Multiple Ways to Name a Point
9:17
Examples
9:30
For Any Angle θ, We Can Make an Equivalent Angle
10:44
Negative Values for r
11:58
If r Is Negative, We Go In The Direction Opposite the One That The Angle θ Points Out
12:22
Another Way to Name the Same Point: Add π to θ and Make r Negative
13:44
Converting Between Rectangular and Polar
14:37
Rectangular Way to Name
14:43
Polar Way to Name
14:52
The Rectangular System Must Have a Right Angle Because It's Based on a Rectangle
15:08
Connect Both Systems Through Basic Trigonometry
15:38
Equation to Convert From Polar to Rectangular Coordinate Systems
16:55
Equation to Convert From Rectangular to Polar Coordinate Systems
17:13
Converting to Rectangular is Easy
17:20
Converting to Polar is a Bit Trickier
17:21
Draw Pictures
18:55
Example 1
19:50
Example 2
25:17
Example 3
31:05
Example 4
35:56
Example 5
41:49
Polar Equations & Functions

38m 16s

Intro
0:00
Introduction
0:04
Equations and Functions
1:16
Independent Variable
1:21
Dependent Variable
1:30
Examples
1:46
Always Assume That θ Is In Radians
2:44
Graphing in Polar Coordinates
3:29
Graph is the Same Way We Graph 'Normal' Stuff
3:32
Example
3:52
Graphing in Polar - Example, Cont.
6:45
Tips for Graphing
9:23
Notice Patterns
10:19
Repetition
13:39
Graphing Equations of One Variable
14:39
Converting Coordinate Types
16:16
Use the Same Conversion Formulas From the Previous Lesson
16:23
Interesting Graphs
17:48
Example 1
18:03
Example 2
18:34
Graphing Calculators, Yay!
19:07
Plot Random Things, Alter Equations You Understand, Get a Sense for How Polar Stuff Works
19:11
Check Out the Appendix
19:26
Example 1
21:36
Example 2
28:13
Example 3
34:24
Example 4
35:52
XII. Complex Numbers and Polar Coordinates
Polar Form of Complex Numbers

40m 43s

Intro
0:00
Polar Coordinates
0:49
Rectangular Form
0:52
Polar Form
1:25
R and Theta
1:51
Polar Form Conversion
2:27
R and Theta
2:35
Optimal Values
4:05
Euler's Formula
4:25
Multiplying Two Complex Numbers in Polar Form
6:10
Multiply r's Together and Add Exponents
6:32
Example 1: Convert Rectangular to Polar Form
7:17
Example 2: Convert Polar to Rectangular Form
13:49
Example 3: Multiply Two Complex Numbers
17:28
Extra Example 1: Convert Between Rectangular and Polar Forms
-1
Extra Example 2: Simplify Expression to Polar Form
-2
DeMoivre's Theorem

57m 37s

Intro
0:00
Introduction to DeMoivre's Theorem
0:10
n nth Roots
3:06
DeMoivre's Theorem: Finding nth Roots
3:52
Relation to Unit Circle
6:29
One nth Root for Each Value of k
7:11
Example 1: Convert to Polar Form and Use DeMoivre's Theorem
8:24
Example 2: Find Complex Eighth Roots
15:27
Example 3: Find Complex Roots
27:49
Extra Example 1: Convert to Polar Form and Use DeMoivre's Theorem
-1
Extra Example 2: Find Complex Fourth Roots
-2
XIII. Counting & Probability
Counting

31m 36s

Intro
0:00
Introduction
0:08
Combinatorics
0:56
Definition: Event
1:24
Example
1:50
Visualizing an Event
3:02
Branching line diagram
3:06
Addition Principle
3:40
Example
4:18
Multiplication Principle
5:42
Example
6:24
Pigeonhole Principle
8:06
Example
10:26
Draw Pictures
11:06
Example 1
12:02
Example 2
14:16
Example 3
17:34
Example 4
21:26
Example 5
25:14
Permutations & Combinations

44m 3s

Intro
0:00
Introduction
0:08
Permutation
0:42
Combination
1:10
Towards a Permutation Formula
2:38
How Many Ways Can We Arrange the Letters A, B, C, D, and E?
3:02
Towards a Permutation Formula, cont.
3:34
Factorial Notation
6:56
Symbol Is '!'
6:58
Examples
7:32
Permutation of n Objects
8:44
Permutation of r Objects out of n
9:04
What If We Have More Objects Than We Have Slots to Fit Them Into?
9:46
Permutation of r Objects Out of n, cont.
10:28
Distinguishable Permutations
14:46
What If Not All Of the Objects We're Permuting Are Distinguishable From Each Other?
14:48
Distinguishable Permutations, cont.
17:04
Combinations
19:04
Combinations, cont.
20:56
Example 1
23:10
Example 2
26:16
Example 3
28:28
Example 4
31:52
Example 5
33:58
Example 6
36:34
Probability

36m 58s

Intro
0:00
Introduction
0:06
Definition: Sample Space
1:18
Event = Something Happening
1:20
Sample Space
1:36
Probability of an Event
2:12
Let E Be An Event and S Be The Corresponding Sample Space
2:14
'Equally Likely' Is Important
3:52
Fair and Random
5:26
Interpreting Probability
6:34
How Can We Interpret This Value?
7:24
We Can Represent Probability As a Fraction, a Decimal, Or a Percentage
8:04
One of Multiple Events Occurring
9:52
Mutually Exclusive Events
10:38
What If The Events Are Not Mutually Exclusive?
12:20
Taking the Possibility of Overlap Into Account
13:24
An Event Not Occurring
17:14
Complement of E
17:22
Independent Events
19:36
Independent
19:48
Conditional Events
21:28
What Is The Events Are Not Independent Though?
21:30
Conditional Probability
22:16
Conditional Events, cont.
23:51
Example 1
25:27
Example 2
27:09
Example 3
28:57
Example 4
30:51
Example 5
34:15
XIV. Conic Sections
Parabolas

41m 27s

Intro
0:00
What is a Parabola?
0:20
Definition of a Parabola
0:29
Focus
0:59
Directrix
1:15
Axis of Symmetry
3:08
Vertex
3:33
Minimum or Maximum
3:44
Standard Form
4:59
Horizontal Parabolas
5:08
Vertex Form
5:19
Upward or Downward
5:41
Example: Standard Form
6:06
Graphing Parabolas
8:31
Shifting
8:51
Example: Completing the Square
9:22
Symmetry and Translation
12:18
Example: Graph Parabola
12:40
Latus Rectum
17:13
Length
18:15
Example: Latus Rectum
18:35
Horizontal Parabolas
18:57
Not Functions
20:08
Example: Horizontal Parabola
21:21
Focus and Directrix
24:11
Horizontal
24:48
Example 1: Parabola Standard Form
25:12
Example 2: Graph Parabola
30:00
Example 3: Graph Parabola
33:13
Example 4: Parabola Equation
37:28
Circles

21m 3s

Intro
0:00
What are Circles?
0:08
Example: Equidistant
0:17
Radius
0:32
Equation of a Circle
0:44
Example: Standard Form
1:11
Graphing Circles
1:47
Example: Circle
1:56
Center Not at Origin
3:07
Example: Completing the Square
3:51
Example 1: Equation of Circle
6:44
Example 2: Center and Radius
11:51
Example 3: Radius
15:08
Example 4: Equation of Circle
16:57
Ellipses

46m 51s

Intro
0:00
What Are Ellipses?
0:11
Foci
0:23
Properties of Ellipses
1:43
Major Axis, Minor Axis
1:47
Center
1:54
Length of Major Axis and Minor Axis
3:21
Standard Form
5:33
Example: Standard Form of Ellipse
6:09
Vertical Major Axis
9:14
Example: Vertical Major Axis
9:46
Graphing Ellipses
12:51
Complete the Square and Symmetry
13:00
Example: Graphing Ellipse
13:16
Equation with Center at (h, k)
19:57
Horizontal and Vertical
20:14
Difference
20:27
Example: Center at (h, k)
20:55
Example 1: Equation of Ellipse
24:05
Example 2: Equation of Ellipse
27:57
Example 3: Equation of Ellipse
32:32
Example 4: Graph Ellipse
38:27
Hyperbolas

38m 15s

Intro
0:00
What are Hyperbolas?
0:12
Two Branches
0:18
Foci
0:38
Properties
2:00
Transverse Axis and Conjugate Axis
2:06
Vertices
2:46
Length of Transverse Axis
3:14
Distance Between Foci
3:31
Length of Conjugate Axis
3:38
Standard Form
5:45
Vertex Location
6:36
Known Points
6:52
Vertical Transverse Axis
7:26
Vertex Location
7:50
Asymptotes
8:36
Vertex Location
8:56
Rectangle
9:28
Diagonals
10:29
Graphing Hyperbolas
12:58
Example: Hyperbola
13:16
Equation with Center at (h, k)
16:32
Example: Center at (h, k)
17:21
Example 1: Equation of Hyperbola
19:20
Example 2: Equation of Hyperbola
22:48
Example 3: Graph Hyperbola
26:05
Example 4: Equation of Hyperbola
36:29
Conic Sections

18m 43s

Intro
0:00
Conic Sections
0:16
Double Cone Sections
0:24
Standard Form
1:27
General Form
1:37
Identify Conic Sections
2:16
B = 0
2:50
X and Y
3:22
Identify Conic Sections, Cont.
4:46
Parabola
5:17
Circle
5:51
Ellipse
6:31
Hyperbola
7:10
Example 1: Identify Conic Section
8:01
Example 2: Identify Conic Section
11:03
Example 3: Identify Conic Section
11:38
Example 4: Identify Conic Section
14:50
XV. Sequences, Series, & Induction
Introduction to Sequences

57m 45s

Intro
0:00
Introduction
0:06
Definition: Sequence
0:28
Infinite Sequence
2:08
Finite Sequence
2:22
Length
2:58
Formula for the nth Term
3:22
Defining a Sequence Recursively
5:54
Initial Term
7:58
Sequences and Patterns
10:40
First, Identify a Pattern
12:52
How to Get From One Term to the Next
17:38
Tips for Finding Patterns
19:52
More Tips for Finding Patterns
24:14
Even More Tips
26:50
Example 1
30:32
Example 2
34:54
Fibonacci Sequence
34:55
Example 3
38:40
Example 4
45:02
Example 5
49:26
Example 6
51:54
Introduction to Series

40m 27s

Intro
0:00
Introduction
0:06
Definition: Series
1:20
Why We Need Notation
2:48
Simga Notation (AKA Summation Notation)
4:44
Thing Being Summed
5:42
Index of Summation
6:21
Lower Limit of Summation
7:09
Upper Limit of Summation
7:23
Sigma Notation, Example
7:36
Sigma Notation for Infinite Series
9:08
How to Reindex
10:58
How to Reindex, Expanding
12:56
How to Reindex, Substitution
16:46
Properties of Sums
19:42
Example 1
23:46
Example 2
25:34
Example 3
27:12
Example 4
29:54
Example 5
32:06
Example 6
37:16
Arithmetic Sequences & Series

31m 36s

Intro
0:00
Introduction
0:05
Definition: Arithmetic Sequence
0:47
Common Difference
1:13
Two Examples
1:19
Form for the nth Term
2:14
Recursive Relation
2:33
Towards an Arithmetic Series Formula
5:12
Creating a General Formula
10:09
General Formula for Arithmetic Series
14:23
Example 1
15:46
Example 2
17:37
Example 3
22:21
Example 4
24:09
Example 5
27:14
Geometric Sequences & Series

39m 27s

Intro
0:00
Introduction
0:06
Definition
0:48
Form for the nth Term
2:42
Formula for Geometric Series
5:16
Infinite Geometric Series
11:48
Diverges
13:04
Converges
14:48
Formula for Infinite Geometric Series
16:32
Example 1
20:32
Example 2
22:02
Example 3
26:00
Example 4
30:48
Example 5
34:28
Mathematical Induction

49m 53s

Intro
0:00
Introduction
0:06
Belief Vs. Proof
1:22
A Metaphor for Induction
6:14
The Principle of Mathematical Induction
11:38
Base Case
13:24
Inductive Step
13:30
Inductive Hypothesis
13:52
A Remark on Statements
14:18
Using Mathematical Induction
16:58
Working Example
19:58
Finding Patterns
28:46
Example 1
30:17
Example 2
37:50
Example 3
42:38
The Binomial Theorem

1h 13m 13s

Intro
0:00
Introduction
0:06
We've Learned That a Binomial Is An Expression That Has Two Terms
0:07
Understanding Binomial Coefficients
1:20
Things We Notice
2:24
What Goes In the Blanks?
5:52
Each Blank is Called a Binomial Coefficient
6:18
The Binomial Theorem
6:38
Example
8:10
The Binomial Theorem, cont.
10:46
We Can Also Write This Expression Compactly Using Sigma Notation
12:06
Proof of the Binomial Theorem
13:22
Proving the Binomial Theorem Is Within Our Reach
13:24
Pascal's Triangle
15:12
Pascal's Triangle, cont.
16:12
Diagonal Addition of Terms
16:24
Zeroth Row
18:04
First Row
18:12
Why Do We Care About Pascal's Triangle?
18:50
Pascal's Triangle, Example
19:26
Example 1
21:26
Example 2
24:34
Example 3
28:34
Example 4
32:28
Example 5
37:12
Time for the Fireworks!
43:38
Proof of the Binomial Theorem
43:44
We'll Prove This By Induction
44:04
Proof (By Induction)
46:36
Proof, Base Case
47:00
Proof, Inductive Step - Notation Discussion
49:22
Induction Step
49:24
Proof, Inductive Step - Setting Up
52:26
Induction Hypothesis
52:34
What We What To Show
52:44
Proof, Inductive Step - Start
54:18
Proof, Inductive Step - Middle
55:38
Expand Sigma Notations
55:48
Proof, Inductive Step - Middle, cont.
58:40
Proof, Inductive Step - Checking In
1:01:08
Let's Check In With Our Original Goal
1:01:12
Want to Show
1:01:18
Lemma - A Mini Theorem
1:02:18
Proof, Inductive Step - Lemma
1:02:52
Proof of Lemma: Let's Investigate the Left Side
1:03:08
Proof, Inductive Step - Nearly There
1:07:54
Proof, Inductive Step - End!
1:09:18
Proof, Inductive Step - End!, cont.
1:11:01
XVI. Preview of Calculus
Idea of a Limit

40m 22s

Intro
0:00
Introduction
0:05
Motivating Example
1:26
Fuzzy Notion of a Limit
3:38
Limit is the Vertical Location a Function is Headed Towards
3:44
Limit is What the Function Output is Going to Be
4:15
Limit Notation
4:33
Exploring Limits - 'Ordinary' Function
5:26
Test Out
5:27
Graphing, We See The Answer Is What We Would Expect
5:44
Exploring Limits - Piecewise Function
6:45
If We Modify the Function a Bit
6:49
Exploring Limits - A Visual Conception
10:08
Definition of a Limit
12:07
If f(x) Becomes Arbitrarily Close to Some Number L as x Approaches Some Number c, Then the Limit of f(x) As a Approaches c is L.
12:09
We Are Not Concerned with f(x) at x=c
12:49
We Are Considering x Approaching From All Directions, Not Just One Side
13:10
Limits Do Not Always Exist
15:47
Finding Limits
19:49
Graphs
19:52
Tables
21:48
Precise Methods
24:53
Example 1
26:06
Example 2
27:39
Example 3
30:51
Example 4
33:11
Example 5
37:07
Formal Definition of a Limit

57m 11s

Intro
0:00
Introduction
0:06
New Greek Letters
2:42
Delta
3:14
Epsilon
3:46
Sometimes Called the Epsilon-Delta Definition of a Limit
3:56
Formal Definition of a Limit
4:22
What does it MEAN!?!?
5:00
The Groundwork
5:38
Set Up the Limit
5:39
The Function is Defined Over Some Portion of the Reals
5:58
The Horizontal Location is the Value the Limit Will Approach
6:28
The Vertical Location L is Where the Limit Goes To
7:00
The Epsilon-Delta Part
7:26
The Hard Part is the Second Part of the Definition
7:30
Second Half of Definition
10:04
Restrictions on the Allowed x Values
10:28
The Epsilon-Delta Part, cont.
13:34
Sherlock Holmes and Dr. Watson
15:08
The Adventure of the Delta-Epsilon Limit
15:16
Setting
15:18
We Begin By Setting Up the Game As Follows
15:52
The Adventure of the Delta-Epsilon, cont.
17:24
This Game is About Limits
17:46
What If I Try Larger?
19:39
Technically, You Haven't Proven the Limit
20:53
Here is the Method
21:18
What We Should Concern Ourselves With
22:20
Investigate the Left Sides of the Expressions
25:24
We Can Create the Following Inequalities
28:08
Finally…
28:50
Nothing Like a Good Proof to Develop the Appetite
30:42
Example 1
31:02
Example 1, cont.
36:26
Example 2
41:46
Example 2, cont.
47:50
Finding Limits

32m 40s

Intro
0:00
Introduction
0:08
Method - 'Normal' Functions
2:04
The Easiest Limits to Find
2:06
It Does Not 'Break'
2:18
It Is Not Piecewise
2:26
Method - 'Normal' Functions, Example
3:38
Method - 'Normal' Functions, cont.
4:54
The Functions We're Used to Working With Go Where We Expect Them To Go
5:22
A Limit is About Figuring Out Where a Function is 'Headed'
5:42
Method - Canceling Factors
7:18
One Weird Thing That Often Happens is Dividing By 0
7:26
Method - Canceling Factors, cont.
8:16
Notice That The Two Functions Are Identical With the Exception of x=0
8:20
Method - Canceling Factors, cont.
10:00
Example
10:52
Method - Rationalization
12:04
Rationalizing a Portion of Some Fraction
12:05
Conjugate
12:26
Method - Rationalization, cont.
13:14
Example
13:50
Method - Piecewise
16:28
The Limits of Piecewise Functions
16:30
Example 1
17:42
Example 2
18:44
Example 3
20:20
Example 4
22:24
Example 5
24:24
Example 6
27:12
Continuity & One-Sided Limits

32m 43s

Intro
0:00
Introduction
0:06
Motivating Example
0:56
Continuity - Idea
2:14
Continuous Function
2:18
All Parts of Function Are Connected
2:28
Function's Graph Can Be Drawn Without Lifting Pencil
2:36
There Are No Breaks or Holes in Graph
2:56
Continuity - Idea, cont.
3:38
We Can Interpret the Break in the Continuity of f(x) as an Issue With the Function 'Jumping'
3:52
Continuity - Definition
5:16
A Break in Continuity is Caused By the Limit Not Matching Up With What the Function Does
5:18
Discontinuous
6:02
Discontinuity
6:10
Continuity and 'Normal' Functions
6:48
Return of the Motivating Example
8:14
One-Sided Limit
8:48
One-Sided Limit - Definition
9:16
Only Considers One Side
9:20
Be Careful to Keep Track of Which Symbol Goes With Which Side
10:06
One-Sided Limit - Example
10:50
There Does Not Necessarily Need to Be a Connection Between Left or Right Side Limits
11:16
Normal Limits and One-Sided Limits
12:08
Limits of Piecewise Functions
14:12
'Breakover' Points
14:22
We Find the Limit of a Piecewise Function By Checking If the Left and Right Side Limits Agree With Each Other
15:34
Example 1
16:40
Example 2
18:54
Example 3
22:00
Example 4
26:36
Limits at Infinity & Limits of Sequences

32m 49s

Intro
0:00
Introduction
0:06
Definition: Limit of a Function at Infinity
1:44
A Limit at Infinity Works Very Similarly to How a Normal Limit Works
2:38
Evaluating Limits at Infinity
4:08
Rational Functions
4:17
Examples
4:30
For a Rational Function, the Question Boils Down to Comparing the Long Term Growth Rates of the Numerator and Denominator
5:22
There are Three Possibilities
6:36
Evaluating Limits at Infinity, cont.
8:08
Does the Function Grow Without Bound? Will It 'Settle Down' Over Time?
10:06
Two Good Ways to Think About This
10:26
Limit of a Sequence
12:20
What Value Does the Sequence Tend to Do in the Long-Run?
12:41
The Limit of a Sequence is Very Similar to the Limit of a Function at Infinity
12:52
Numerical Evaluation
14:16
Numerically: Plug in Numbers and See What Comes Out
14:24
Example 1
16:42
Example 2
21:00
Example 3
22:08
Example 4
26:14
Example 5
28:10
Example 6
31:06
Instantaneous Slope & Tangents (Derivatives)

51m 13s

Intro
0:00
Introduction
0:08
The Derivative of a Function Gives Us a Way to Talk About 'How Fast' the Function If Changing
0:16
Instantaneous Slop
0:22
Instantaneous Rate of Change
0:28
Slope
1:24
The Vertical Change Divided by the Horizontal
1:40
Idea of Instantaneous Slope
2:10
What If We Wanted to Apply the Idea of Slope to a Non-Line?
2:14
Tangent to a Circle
3:52
What is the Tangent Line for a Circle?
4:42
Tangent to a Curve
5:20
Towards a Derivative - Average Slope
6:36
Towards a Derivative - Average Slope, cont.
8:20
An Approximation
11:24
Towards a Derivative - General Form
13:18
Towards a Derivative - General Form, cont.
16:46
An h Grows Smaller, Our Slope Approximation Becomes Better
18:44
Towards a Derivative - Limits!
20:04
Towards a Derivative - Limits!, cont.
22:08
We Want to Show the Slope at x=1
22:34
Towards a Derivative - Checking Our Slope
23:12
Definition of the Derivative
23:54
Derivative: A Way to Find the Instantaneous Slope of a Function at Any Point
23:58
Differentiation
24:54
Notation for the Derivative
25:58
The Derivative is a Very Important Idea In Calculus
26:04
The Important Idea
27:34
Why Did We Learn the Formal Definition to Find a Derivative?
28:18
Example 1
30:50
Example 2
36:06
Example 3
40:24
The Power Rule
44:16
Makes It Easier to Find the Derivative of a Function
44:24
Examples
45:04
n Is Any Constant Number
45:46
Example 4
46:26
Area Under a Curve (Integrals)

45m 26s

Intro
0:00
Introduction
0:06
Integral
0:12
Idea of Area Under a Curve
1:18
Approximation by Rectangles
2:12
The Easiest Way to Find Area is With a Rectangle
2:18
Various Methods for Choosing Rectangles
4:30
Rectangle Method - Left-Most Point
5:12
The Left-Most Point
5:16
Rectangle Method - Right-Most Point
5:58
The Right-Most Point
6:00
Rectangle Method - Mid-Point
6:42
Horizontal Mid-Point
6:48
Rectangle Method - Maximum (Upper Sum)
7:34
Maximum Height
7:40
Rectangle Method - Minimum
8:54
Minimum Height
9:02
Evaluating the Area Approximation
10:08
Split the Interval Into n Sub-Intervals
10:30
More Rectangles, Better Approximation
12:14
The More We Us , the Better Our Approximation Becomes
12:16
Our Approximation Becomes More Accurate as the Number of Rectangles n Goes Off to Infinity
12:44
Finding Area with a Limit
13:08
If This Limit Exists, It Is Called the Integral From a to b
14:08
The Process of Finding Integrals is Called Integration
14:22
The Big Reveal
14:40
The Integral is Based on the Antiderivative
14:46
The Big Reveal - Wait, Why?
16:28
The Rate of Change for the Area is Based on the Height of the Function
16:50
Height is the Derivative of Area, So Area is Based on the Antiderivative of Height
17:50
Example 1
19:06
Example 2
22:48
Example 3
29:06
Example 3, cont.
35:14
Example 4
40:14
XVII. Appendix: Graphing Calculators
Buying a Graphing Calculator

10m 41s

Intro
0:00
Should You Buy?
0:06
Should I Get a Graphing Utility?
0:20
Free Graphing Utilities - Web Based
0:38
Personal Favorite: Desmos
0:58
Free Graphing Utilities - Offline Programs
1:18
GeoGebra
1:31
Microsoft Mathematics
1:50
Grapher
2:18
Other Graphing Utilities - Tablet/Phone
2:48
Should You Buy a Graphing Calculator?
3:22
The Only Real Downside
4:10
Deciding on Buying
4:20
If You Plan on Continuing in Math and/or Science
4:26
If Money is Not Particularly Tight for You
4:32
If You Don't Plan to Continue in Math and Science
5:02
If You Do Plan to Continue and Money Is Tight
5:28
Which to Buy
5:44
Which Graphing Calculator is Best?
5:46
Too Many Factors
5:54
Do Your Research
6:12
The Old Standby
7:10
TI-83 (Plus)
7:16
TI-84 (Plus)
7:18
Tips for Purchasing
9:17
Buy Online
9:19
Buy Used
9:35
Ask Around
10:09
Graphing Calculator Basics

10m 51s

Intro
0:00
Read the Manual
0:06
Skim It
0:20
Play Around and Experiment
0:34
Syntax
0:40
Definition of Syntax in English and Math
0:46
Pay Careful Attention to Your Syntax When Working With a Calculator
2:08
Make Sure You Use Parentheses to Indicate the Proper Order of Operations
2:16
Think About the Results
3:54
Settings
4:58
You'll Almost Never Need to Change the Settings on Your Calculator
5:00
Tell Calculator In Settings Whether the Angles Are In Radians or Degrees
5:26
Graphing Mode
6:32
Error Messages
7:10
Don't Panic
7:11
Internet Search
7:32
So Many Things
8:14
More Powerful Than You Realize
8:18
Other Things Your Graphing Calculator Can Do
8:24
Playing Around
9:16
Graphing Functions, Window Settings, & Table of Values

10m 38s

Intro
0:00
Graphing Functions
0:18
Graphing Calculator Expects the Variable to Be x
0:28
Syntax
0:58
The Syntax We Choose Will Affect How the Function Graphs
1:00
Use Parentheses
1:26
The Viewing Window
2:00
One of the Most Important Ideas When Graphing Is To Think About The Viewing Window
2:01
For Example
2:30
The Viewing Window, cont.
2:36
Window Settings
3:24
Manually Choose Window Settings
4:20
x Min
4:40
x Max
4:42
y Min
4:44
y Max
4:46
Changing the x Scale or y Scale
5:08
Window Settings, cont.
5:44
Table of Values
7:38
Allows You to Quickly Churn Out Values for Various Inputs
7:42
For example
7:44
Changing the Independent Variable From 'Automatic' to 'Ask'
8:50
Finding Points of Interest

9m 45s

Intro
0:00
Points of Interest
0:06
Interesting Points on the Graph
0:11
Roots/Zeros (Zero)
0:18
Relative Minimums (Min)
0:26
Relative Maximums (Max)
0:32
Intersections (Intersection)
0:38
Finding Points of Interest - Process
1:48
Graph the Function
1:49
Adjust Viewing Window
2:12
Choose Point of Interest Type
2:54
Identify Where Search Should Occur
3:04
Give a Guess
3:36
Get Result
4:06
Advanced Technique: Arbitrary Solving
5:10
Find Out What Input Value Causes a Certain Output
5:12
For Example
5:24
Advanced Technique: Calculus
7:18
Derivative
7:22
Integral
7:30
But How Do You Show Work?
8:20
Parametric & Polar Graphs

7m 8s

Intro
0:00
Change Graph Type
0:08
Located in General 'Settings'
0:16
Graphing in Parametric
1:06
Set Up Both Horizontal Function and Vertical Function
1:08
For Example
2:04
Graphing in Polar
4:00
For Example
4:28
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Lecture Comments (8)

2 answers

Last reply by: Peter Ke
Wed Nov 18, 2015 6:59 PM

Post by Peter Ke on November 14, 2015

I have a question about Example 4. I understand what you did but when I add the answer which was 109.2N + 244.6N = 353.8N which is not exactly equal to 300N.

Because you said that the net force must equal <0,0>, so I was just wondering why it's about 50N greater than 300N?

0 answers

Post by Joshua Jacob on July 25, 2014

I liked the pace of the lectures! It forced me to go back and fully understand each idea. Thank you for doing an awesome job!

1 answer

Last reply by: Professor Selhorst-Jones
Sun Apr 20, 2014 8:43 PM

Post by Tim Zhang on April 15, 2014

This lecture is so amazing and fantastic!!!!!!!! good job. by the way ,I like the fast speed.

1 answer

Last reply by: Professor Selhorst-Jones
Fri Aug 30, 2013 2:50 PM

Post by Edwin Wong on August 30, 2013

You go to fast with the lectures. Please slow down next time

Vectors

  • A vector is a way to talk about magnitude (length/size) and direction (angle) at the same time.
  • We can visualize a vector as a directed line segment-a length with direction.
  • Vectors are normally denoted by an overhead arrow u  or put in bold face u. [And once in a while it's just assumed: u.] Vectors are usually named with lowercase letters, but just like variables, any symbol can be used in theory. Most often you will see u and v.
  • If we know the vector's location in the plane, we can give the vector algebraically in component form. For example, u = 〈3,4〉. We normally use angle brackets 〈,  〉 to denote vectors, but we will often also see parentheses (,   ). [After we learn about unit vectors, we'll also talk about using i, j.]
  • We denote the magnitude (length) of a vector u by  |u|. [Also sometimes written as ||u|| .] If we know the component form of a vector, we can figure out its length by the Pythagorean theorem. This gives the formula
    |

    u
     

     
    |  = | 〈a, b 〉|   =   

     

    a2 + b2
     

     
    .
  • If we know the component form of a vector, we can figure out its angle through trigonometry. We normally talk about direction as the counter-clockwise angle from the positive x-axis (just like the unit circle), but sometimes our reference location changes. In any case, always draw a picture before trying to figure out or use angles.
  • If we know the magnitude and angle of a vector, we can figure out its component form. First draw a sketch to help you see it, then use trigonometry.
  • If we have a vector u, we can scale it to a different length by multiplying by a scalar: a real number. Algebraically, the scalar just multiplies each component. In general, for a scalar k and vector u = 〈a, b 〉,
    k

    u
     

     
    = k ·〈a, b 〉 = 〈ka, kb 〉.
  • A unit vector is a vector with a length of 1. It still has a direction, but it must have a magnitude of exactly 1. We can create a unit vector out of any vector u by dividing u by its length, |u|.
    Unit vector:  

    u
     

     

    |

    u
     

     
    |



     
    Note that the above has the same direction as u, but is length 1.
  • We can combine two (or more) vectors through addition (or subtraction). We simply add them component-wise: the horizontal components from each vector combine, as do the vertical components.

    u
     

     
    +

    v
     

     
    = 〈u1, u2 〉+ 〈v1, v2 〉 = 〈u1 + v1,   u2 + v2
  • We can see this idea of combining vectors geometrically as well. Adding vectors is done by placing the tail of one at the head of the other. Then we draw in a vector from the original start to the final end.
  • We can put together the idea of combining vectors with the idea of unit vectors to get a new way to express a vector's components. We start by creating two standard unit vectors, one horizontal and one vertical:
    i = 〈1, 0 〉,                      j = 〈0, 1 〉.
    With this, we can express any vector in terms of i and j:
    〈3, 4 〉 = 3 i + 4 j,               〈4.7, π〉 = 4.7 i + πj.
  • We can also talk about a zero vector, denoted 0, that has 0's in all its components. It has no length (and so its direction does not matter).
  • While vectors have many operations that are similar to things we are already used to, there is no analogue to multiplication. There is no good way to define multiplying two vectors, so we do not define vector multiplication. [Still, there is an operation somewhat similar to multiplication called the dot product. We'll explore that and what it means in the next lesson.]
  • Motion in a medium is the combination of the object's motion vector relative to the medium and the medium's motion vector:

    v
     

     
    +

    v
     

     
      =  

    v
     

     
    .
  • Vectors can work in arbitrarily high dimensions. While all of the above was only done in two dimensions, a vector can have any number of components. By the way we defined scalars and vector combination, everything we've discussed so far about vectors still works fine. They might get confusing to picture in higher dimensions, but everything still makes sense. [In three dimensions, we have another standard unit vector:
    k = 〈0, 0 , 1 〉           (and in three dimensions  i = 〈1,0,0〉,     j = 〈0,1,0 〉)
  • For any n-dimensional vector

    x
     

     
    = 〈x1,  x2, …,  xn 〉,
    the length of the vector is simply the square root of the sums of each of its components squared:
    |

    x
     

     
    | =

     

    x1  2 + x2  2 + …+ xn  2
     

     
    .

Vectors

Given that u = 〈4,  5 〉,  v = 〈2,  −3 〉,  and w = 〈−2,  0 〉, find the below.

u
 
+

v
 
                    

u
 
+

v
 
+

w
 
  • When adding vectors, we add component-wise, that is, we add the first components together to make a new first component, we add the second components together to make a new second component, and so on.
  • For the first expression, we have

    u
     
    +

    v
     
        =     〈4,  5 〉+ 〈2,  −3 〉    =     〈4+2,    5 −3 〉    =     〈6,  2 〉
  • For the second expression, we have

    u
     
    +

    v
     
    +

    w
     
        =     〈4,  5 〉+ 〈2,  −3 〉+ 〈−2,  0 〉    =     〈4+2−2,    5 −3 +0 〉    =     〈4,  2 〉
u+v = 〈6,  2 〉,       u + v+w = 〈4,  2 〉
Given that u = 〈−7,  3 〉 and v = 〈5,  10 〉, find the below.
8

u
 
                     − 1

2

v
 
  • The number in front of the vector is called a scalar. When a scalar is in front of a vector, it multiplies the vector. Applying a scalar to a vector is as simple as multiplying each of the components by the scalar (like distribution).
  • For the first expression, we have
    8

    u
     
        =     8 ·〈−7,  3 〉    =     〈8 ·(−7),    8·3 〉    =     〈−56,  24 〉
  • For the second expression, we have
    1

    2

    v
     
        =     − 1

    2
    ·〈5,  10 〉    =     
     

    1

    2

    ·5,    
    1

    2

    ·10 
     
        =     
     
    5

    2
    ,  − 5 
     
8 u = 〈−56,  24 〉,        − [1/2] v = 〈 −[5/2],  − 5 〉
Given that u = 〈1,  2 〉,  v = 〈3,  −1 〉,  and w = 〈−2,  −2 〉, find the below.
5

u
 
−3

v
 
                     3(

u
 
+ 2

v
 
) −5

w
 
  • Vector addition (or subtraction) is done component-wise: each portion of the vector only interacts with the equivalent portion in the other vector. Scalar multiplication (a normal, real number in front of the vector) is the same as distributing that number through multiplication.
  • Start off by applying the scalars to their vectors, then add the results together. Here's the first expression:
    5

    u
     
    −3

    v
     
        =     5 ·〈1,  2 〉− 3 ·〈3,  −1 〉    =     〈5,  10 〉+ 〈−9,  3 〉    =     〈−4,  13 〉
  • We work similarly with the second expression, but pay attention to the parentheses-they mean the same thing as we're used to. Expressions inside of a parentheses get priority, but they can also be distributed on to.
    3(

    u
     
    + 2

    v
     
    ) −5

    w
     
        =     3 (  〈1,  2 〉+ 2·〈3,  −1 〉 ) −5 ·〈−2,  −2 〉    =     3(  〈1,  2 〉+ 〈6,  −2 〉 ) + 〈10,  10 〉

           = 3(  〈7,  0 〉 ) + 〈10,  10 〉    =     〈21,  0 〉+ 〈10,  10 〉    =     〈31,  10〉
    [Alternatively, instead of figuring out the vector inside of the parentheses first, we could distribute right at the start:
    3(

    u
     
    + 2

    v
     
    ) −5

    w
     
        =     3

    u
     
    + 6

    v
     
    − 5

    w
     
        =     3 〈1,  2 〉+ 6 〈3,  −1 〉−5 〈−2,  −2 〉    =     〈31,  10 〉
    Both methods work fine, and they will give the same result. Use whichever you feel more comfortable with.]
5u −3v = 〈−4,  13 〉,        3(u + 2 v) −5 w = 〈31,  10 〉
Let u = 〈3,  4 〉 and v = 〈−10,  −2 〉. Show how we can geometrically find the resultant vector of the below expression.
2

u
 
+

v
 
  • To draw the first vector of u have the tail start at the origin and the head of the vector will fall on the point (3,  4), since the vector is 〈3,  4 〉.
  • Next, we need to apply the scalar of 2 to u. The scalar scales the vector by that amount. In this case, we scale it by a factor of 2, so we will double the length of the vector. This means we will now draw from the origin out twice as far to (6,  8).
  • Finally, we need to add the vector v. Adding a vector geometrically is simple: instead of having the second vector also "start" at the origin, it starts from where the first vector "ended" (the first vector's head). Thus, we will place v so that it's tail is at the point (6,  8), and it will travel 〈−10,  −2 〉 to have its head at the point (−4,  6). The resultant vector of the expression 2u + v has its tail at the origin and its head goes to that same final point.
  • We can check our geometric work that we did graphically with algebra. If we did this algebraically, we would have
    2

    u
     
    +

    v
     
        =     2 ·〈3,  4 〉+ 〈−10,  −2 〉    =     〈6,  8 〉+ 〈−10,  −2 〉    =     〈−4,  6 〉
    This is exactly what we got from figuring it out geometrically, as you can see in the picture for the answer.
Find the magnitude and angle of the below vector.
〈−2,  −5 〉
  • Finding the magnitude of any vector is quite easy: just take the square root of the sum of each component squared and added together.
    | 〈−2,  −5 〉|     =    

     

    (−2)2+(−5)2
     
        =    

     

    29
     
  • Finding the angle that the vector has is not much harder, but it is very important to make a diagram. If you don't make a diagram to help yourself see what's going on, you might make a mistake. We want to find the angle θ below:
  • We can use trigonometry to solve for the angle that the vector makes, but it will not directly give us the value of θ. None of the inverse trigonometric functions can output an angle of that size. Instead, the safest way to do this problem is to ask ourselves, "What angle does the vector make with the horizontal axis?" Thus, we want to know the value of the green `?' symbol in the below diagram. To help us do this, we can treat the vector as a right triangle, and put down the length of its two legs.
  • We can find the value of the `?' angle by simple right triangle trigonometry:
    tan (?) = 5

    2
        ⇒     ? = tan−1
    5

    2

    Plugging that in to a calculator, we get that    ?=68.199°. Remember though, our ultimate goal is to find θ. Notice that to make θ, it has to go over the entire top two quadrants (180°), then add on the angle of `?'. Therefore, the angle of θ is given by
    θ    =     ? + 180     =     68.199 + 180     =     248.199
    Whenever you need to find the angle of a vector, make sure to draw a diagram first. It's extremely useful to have a visual reference so you can make sense of the angle you're looking for.
Magnitude: √{29},     Angle: 248.199°
The angle u has a magnitude of |u| = 19 and angle of θ = 147°. Give the component form of u. (Round your answer to three decimal places.)
  • Finding the component form of a vector if we know the magnitude (length) and angle is quite easy: we simply use trigonometry. The cosine function allows us to find the x-value and the sine function allows us to find the y-value. If you have difficulty understanding how the below steps work, try drawing a sketch of the vector, then create a right triangle where the vector, the x portion, and the y portion make the three sides.
  • The distance form the origin is the magnitude of the vector, and we know the angle that the vector is on. Thus, if we want to find the horizontal x component, we have
    cosθ = x

    r
        ⇒     cos147° = x

    19
        ⇒     19 ·cos147° = x
    Plugging into a calculator, we get that x = −15.935. [If you get a different value on your calculator, make sure your calculator is set to degrees mode and not radians.]
  • Working similarly to find the vertical y component, we have
    sinθ = y

    r
        ⇒     sin147° = y

    19
        ⇒     19 ·sin147° = y
    Plugging into a calculator, we get that y = 10.348.
  • Finally, because the vector u is comprised of both components, we have to put them together into a single vector:

    u
     
        =     〈x,  y 〉    =     〈−15.935,   10.348 〉
u = 〈−15.935,   10.348 〉
Find the unit vector pointing in the same direction as the vector u = 〈36,  77 〉.
  • A unit vector is a vector with magnitude 1. We can create a unit vector out of any vector (that is, create a vector that points in the exact same direction, but now has a length of 1). We do this by finding the magnitude of the vector, then dividing the vector by that.
    Unit vector: 

    u

    |

    u
     
    |
    Since [1/(|u|)] is just a scalar, this scales the vector to a length of 1, no matter what length u started at.
  • Begin by finding |u|:
    |

    u
     
    |     =    

     

    362 + 772
     
        =     85
  • Once you know the magnitude, divide the original vector by it:
    Unit vector: 

    u

    |

    u
     
    |
        =    〈36,  77 〉

    85
        =     
     
    36

    85
    ,   77

    85
     
     
Unit vector in same direction as u: 〈 [36/85],  [77/85] 〉
A large boat is being pulled along by two tugboats. Tugboat A is pulling along the boat with a force of 7000 N (N →newtons, the metric unit for force) and at an angle of 30° north of east. Tugboat B is pulling the boat with a force of 5000 N and at an angle of 50° south of east. What is the resultant force? (Give your answer in component form.) [The resultant force is the sum of the forces acting upon an object.]
  • We can express the force exerted by each tugboat as a vector. Once we know the force vector of each tugboat, we can sum them together to find the combined effect of both pulls-the resultant force.
  • Let us consider east and west as being the positive and negative x-directions respectively. Similarly, let us consider north and south as being the positive and negative y-directions respectively. With this set up, we can find the component form of each tugboat's force vector by using trigonometry. Tugboat A:
    cos(30°) = x

    7000
        ⇒     x = 6062.2       
           sin(30°) = y

    7000
        ⇒     y = 3500
    Thus tugboat A has a force vector of FA=〈6062.2,   3500 〉. Tugboat B:
    cos(−50°) = x

    5000
        ⇒     x = 3213.9       
           sin(−50°) = y

    5000
        ⇒     y = −3830.2
    Thus tugboat B has a force vector of FB = 〈3213.9,   −3830.2 〉.
  • To find the resultant force, we simply add together the force vectors for each tugboat on its own:

    F
     

    A 
    +

    F
     

    B 
        =     〈6062.2,   3500 〉+ 〈3213.9,   −3830.2 〉    =     〈9276.1,   −330.2 〉
〈9276.1 N,   −330.2 N 〉
A crate weighing 700 N is suspended from a pair of cables A and B, as in the diagram. Using the diagram, figure out the magnitude of the tension in each cable.
  • To do this problem, we must realize a deceptively simple fact: the crate is not moving. If the crate is not moving, that means the total force on the crate must come out to nothing. Since we can express force as a vector, that means the total force on the crate must be the zero vector:
    Total force on crate: 〈0,  0 〉
  • The total force on the crate is the sum of all the other forces being applied to the crate. Let us call the force applied by cable A the vector A. Similarly, we can call the force applied by cable B the vector B. Notice that gravity is also pulling down on the crate (in the form of its weight); let's call that g for right now. Therefore, the sum of these three force vectors must result in the total force that we just talked about:

    A
     
    +

    B
     
    +

    g
     
    = 〈0,  0 〉
  • While we can't figure out the component form of A or B (yet), we can connect the magnitude (length/hypotenuse) of each vector to its components through trigonometry. For cable A, we can find the length of the horizontal component as
    cos(52°) = xA

    |

    A
     
    |
        ⇒     xA = |

    A
     
    | cos(52°)
    Similarly, the length of the vertical component is
    sin(52°) = yA

    |

    A
     
    |
        ⇒     yA = |

    A
     
    | sin(52°)
    From this we can create the component form of the vector A. However!, there is one slight issue. Notice that the cable will pull to the left on the crate. Thus, instead of a positive value for the x-component of A, it must be a negative value, since it pulls to the left. Put this all together:

    A
     
    = 〈− |

    A
     
    | cos(52°),     |

    A
     
    | sin(52°) 〉


    We can do a very similar thing to find B (although its x component will be positive, since it pulls the crate to the right):

    B
     
    = 〈|

    B
     
    | cos(39°),     |

    B
     
    | sin(39°) 〉
  • We can also figure out the component form for the force of gravity. Gravity only pulls down, so all of the weight is pulling in the y-direction, none in the x. Thus, for gravity, we have

    g
     
    = 〈0,  −700 〉.
    From earlier, we know

    A
     
    +

    B
     
    +

    g
     
    = 〈0,  0 〉,
    so we can plug in the vectors we now know:
    〈− |

    A
     
    | cos(52°),   |

    A
     
    | sin(52°) 〉+ 〈|

    B
     
    | cos(39°),   |

    B
     
    | sin(39°) 〉+ 〈0,  −700 〉 = 〈0,  0 〉
    That means we know that the sum of each component type comes out to be 0.
  • Doing this for the x-components first, we get
    − |

    A
     
    | cos(52°) + |

    B
     
    | cos(39°) + 0 = 0     ⇒     |

    B
     
    | cos(39°) = |

    A
     
    | cos(52°)
    Doing this with the y-components next, we get
    |

    A
     
    | sin(52°) + |

    B
     
    | sin(39°) − 700 = 0     ⇒     |

    A
     
    | sin(52°) + |

    B
     
    | sin(39°) = 700
  • We now have a system of equations that we can solve by substitution. Solve for one of the vector magnitudes in the first equation, then plug that in to the other one and solve:
    |

    B
     
    | cos(39°) = |

    A
     
    | cos(52°)     ⇒     |

    B
     
    | = |

    A
     
    | · cos(52°)

    cos(39°)
    Plugging in:
    |

    A
     
    | sin(52°) +
    |

    A
     
    | · cos(52°)

    cos(39°)

    sin(39°) = 700

    |

    A
     
    | ·
    sin(52°) + cos(52°) ·sin(39°)

    cos(39°)

    = 700

    |

    A
     
    | = 700

    sin(52°) + cos(52°) ·sin(39°)

    cos(39°)
        =     544.09


    Now that we know |A| = 544.09, we can plug in to find |B|:
    |

    B
     
    | = (544.09) · cos(52°)

    cos(39°)
        =     431.03
Cable A's magnitude: 544.09 N,    Cable B's magnitude: 431.03 N
A boat is in the water and needs to go directly north at 10 m/s. However, the water is flowing from the NE to the SW at an angle of 20° south of west and a speed of 2 m/s. What speed and angle must the boat have relative to the water to achieve its goal velocity?
  • We will approach this problem based on the idea of motion in a medium. The total motion of an object is affected by the motion of the medium and the object's motion relative to the medium it is in. Mathematically, using vectors, we have

    v
     

    object 
    +

    v
     

    medium 
      =  

    v
     

    total motion 
    .
    For this problem, the motion of the water combined with the motion of the ship relative to the water (what the problem asked for) must equal the ship's goal velocity.
  • To continue, we need to put everything in vector component form so we can add things together. The goal velocity of the ship is going entirely to the north at 10 m/s, so we can write that as
    Total motion:    〈0,  10 〉
  • Next, we figure out the velocity vector for the water. Using trig, we can find the length of the horizontal (xw) and vertical (yw) components:
    cos(20°) = xw

    2
        ⇒     xw = 1.879       
           sin(20°) = yw

    2
        ⇒     yw = 0.684
    We can now put this together to find the velocity vector for the water. However!, we must pay attention to our diagram. The water's motion is moving to the left (west) and down (south). This means that the horizontal and negative components must be negative. While the lengths are the above, we have to pay attention and realize that those components will be negative because of their directions.
    Velocity of water:    〈−1.879,  −0.684 〉
  • We don't know anything about the velocity of the boat relative to the water yet, so let's just call it v, and we can name its components as v = 〈xv, yv 〉. With this set up, we can now add together the velocity of the water and the velocity of the boat relative to the water and get the total velocity:
    〈−1.879,  −0.684 〉+ 〈xv, yv 〉 = 〈0,  10 〉
  • Since vectors add component-wise, we don't have to worry about first components affecting second components and vice-versa. Thus we can break this into two separate equations:
    −1.879 + xv = 0           
               −0.684 + yv = 10
    Solve for xv and yv:
    xv = 1.879           
               yv = 10.684
    Therefore, the ship's velocity vector relative to the water must be

    v
     
    = 〈1.879,   10.684 〉
  • Finally, the problem asked for v's angle and speed. Speed just means the magnitude of the velocity vector, so we find that as
    |

    v
     
    | =

     

    1.8792 + 10.6842
     
        =     10.848
    To find the angle, we can just use simple trigonometry:
    tan(θ) = 10.684

    1.879
        ⇒     θ = tan−1
    10.684

    1.879

        =     80.025°
Ship's speed relative to water: 10.848 m/s,    Ship's angle of motion relative to water: 80.025° north of east

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

Vectors

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Introduction 0:10
    • Magnitude of the Force
    • Direction of the Force
    • Vector
  • Idea of a Vector 1:30
    • How Vectors are Denoted
  • Component Form 3:20
    • Angle Brackets and Parentheses
  • Magnitude/Length 4:26
    • Denoting the Magnitude of a Vector
  • Direction/Angle 7:52
    • Always Draw a Picture
  • Component Form from Magnitude & Angle 10:10
  • Scaling by Scalars 14:06
  • Unit Vectors 16:26
  • Combining Vectors - Algebraically 18:10
  • Combining Vectors - Geometrically 19:54
    • Resultant Vector
  • Alternate Component Form: i, j 21:16
  • The Zero Vector 23:18
  • Properties of Vectors 24:20
  • No Multiplication (Between Vectors) 28:30
    • Dot Product
  • Motion in a Medium 30:10
    • Fish in an Aquarium Example
  • More Than Two Dimensions 33:12
  • More Than Two Dimensions - Magnitude 34:18
  • Example 1 35:26
  • Example 2 38:10
  • Example 3 45:48
  • Example 4 50:40
    • Example 4, cont.
  • Example 5 1:01:32

Transcription: Vectors

Hi--welcome back to Educator.com.0000

Today, we are going to talk about vectors; this starts an entirely new section for us.0002

We are getting into really new territory; we are going to talk about vectors, and then later on we will be talking about matrices.0007

But first, let's talk about vectors: when we talk about the force on an object, we need to know two things:0011

the magnitude of the force--how hard it is being pushed, and the direction--which way it is being pushed.0016

If we are going to talk about something being pushed, we can't just talk about how hard it is being pushed.0023

We also have to know which way it is being pushed; it is very different to push something this way than it is to push something this way.0027

But even if we know the direction, we also have to know how hard it is.0032

We are pushing really hard versus pushing very slightly--there are big differences here.0034

We need to know both magnitude and direction; we need to know how hard and which way.0039

Now, one number on its own won't be enough to get across both pieces of information.0044

So, this leads us to the idea of a vector--a way to be able to talk about magnitude and direction at the same time--0048

be able to talk about both of these things in one piece of information.0054

Vectors are massively useful: they are used throughout math; they are in the sciences everywhere, especially in physics;0058

they are in engineering, computer programming, business, medicine, and more fields.0065

Pretty much, if a field even vaguely uses math, it is going to use vectors; so this is a really useful thing that we are talking about in this lesson.0069

Now, I want to point out that this lesson will use some basic trigonometry to figure out angles.0076

So, make sure you have some familiarity with how trigonometry works, so that you can understand what is going on when we are figuring out angles.0080

All right, let's go on: the idea of a vector: a vector is just trying to get across the idea of length and direction.0087

Graphically, if we look at a picture of it, it is exactly that: it is a directed line segment, a length with direction.0095

It is both of these things at once.0102

So, in this case, we are starting at (0,0), and we have this length here; and the arrow at the end says that we are going in this direction.0104

So, we have a chunk of length, and we see which way it is pointed.0112

It is pointed in a very specific direction; it has some angle to it.0116

How do we call out a vector, if we want to talk about a vector?0120

Normally, we denote it with an overhead arrow like this: u with an arrow above it says that we are talking about the vector u.0123

Or we can put it in bold face with a bold u.0131

And once in a while, if we are talking about nothing but vectors, and there is nothing else showing up,0135

sometimes it will just be assumed; but we are pretty much never going to see that--not in this course.0140

Vectors are normally shown with lowercase letters; however, just like variables, you can use any symbol.0145

But for the most part, we will stick with lowercase letters: u and v are very common letters for talking about them.0150

One last thing: we use this u with the arrow on top, and that is how we will be denoting it in this course, although you might see bold face in textbooks.0157

When you are actually writing it by hand, I write it like this, where I don't really have a full arrow.0166

I have more of a harpoon, where it is just an arrow on one side--just this sort of arrow flange on one side.0171

That is how I write it; you could write it with an actual arrow on top--there is nothing wrong with that.0178

But I am lazy, like us all, and so I just tend to write it like that, because it is the fastest way that I know to write it.0182

But it still gets across the idea of a vector.0189

So, this is a perfectly fine way to write it by hand when you are working it out.0191

But when we have it actually written out in this lesson, we will have it like that with an arrow on top.0194

We can also give the vector algebraically by its location in the plane.0200

We have these nice rectangular coordinates; we can break that into components.0204

We call this the component form: we have a horizontal amount of 3 and a vertical amount of 4.0209

So, we say that its horizontal component is 3, and its vertical component is 4.0215

u, the vector u, equals (3,4); the first component is the horizontal one, and the second component is the vertical,0220

just like when we are talking about points in the plane.0227

We normally use angle brackets (these are angle brackets, because they are at an angle) to denote vectors.0230

But we will often also see parentheses; parentheses are very common, as well.0238

I personally actually tend to use parentheses more; but most precalculus/math analysis courses tend to use angle brackets.0241

So, I am teaching with angle brackets; but personally, when I am just doing math on my own, I often tend to use parentheses.0249

But either of them is just fine.0254

After we learn about unit vectors, we will see one more way to talk about the component form, using i and j.0256

But we will leave that until a little bit later, once we have actually talked about unit vectors.0261

If we know the component form of a vector, we can figure out its length by the Pythagorean theorem.0265

Remember: the Pythagorean theorem says that the hypotenuse to a triangle, squared, is equal to both of the legs, squared and then added together.0270

So, that means that we can take the square root of both sides of the equation.0284

We have that the hypotenuse is equal to the square root of each leg, squared and added together, the square root of (leg squared + leg squared).0287

In this case, we know that our vector is (3,4); so we have that one leg is 3, and one leg is 4.0295

So, we work this out; √(32 + 42) becomes √(9 + 16), or √25, so we get 5.0306

So, we see that the length of this vector is 5.0312

When we want to denote the magnitude of a vector, when we want to talk about the length of a vector,0315

the magnitude of a vector, we normally use the word "magnitude" to talk about length.0320

But in either case, with "length" or "magnitude," we are just talking about how long the vector is.0324

We normally use these vertical bars on either side of it, just like we do with absolute value.0329

And we will talk about that in just a moment--why it is like absolute value.0336

Also, sometimes you will see it written ||u||, vector u with double bars around it.0339

In either case, whether it is single bar or single bar, what we are talking about is how long that vector is, if we measure it from the origin out to its tip.0346

If u is equal to (a,b), then the length of u, the bars on either side of u (which would also be bars on either side of (a,b),0354

because that is just a vector, as well), will give us the square root of a2 + b2,0363

based on this exact same reasoning that the length of our hypotenuse is equal to the square root of each leg, squared and added together.0368

So in this case, a and b are just there and there on our vector.0376

Now, I would like to point out, really quickly, why in the world we are using vertical bars on either side,0381

just like we did with talking about absolute value.0385

Let's think about that for a bit: let's think about...when we work with absolute value, say we have 0 here, and here is +5, and here is -5.0389

Well, if we talk about the absolute value of 5, and we talk about the absolute value of -5,0398

in both cases, we are going to get the number 5 out of it.0403

We get 5 in either case, because what the absolute value is telling us is: it is saying how far you are from 0--how far you are from the origin.0408

So, the reason why the absolute value of 5 is 5 is because it takes 5 units of length to get from 0 to 5.0417

And that is the exact same reason why the absolute value of -5 is 5: because it takes 5 units of length to get from 0 to -5.0424

They go in different directions, but it is a question of how far away it is.0431

They are both 5 units away from the origin.0435

The same thing is going on when we are dealing with a vector.0437

When we talk about the size of a vector, what the length of a vector is, we are asking how far the vector goes out from the origin.0440

In both the case of the absolute value and the magnitude of a vector, what we are saying is, "How far are you from the origin?"0450

We put bars around the u because it is basically doing the same thing as bars around a number with absolute value.0457

Bars around a number are saying, "How far are you from 0?"; bars around a vector are saying, "How far are you from the origin?"0463

So, they are both a question of length, effectively.0468

All right, direction and angle: if we know the component form of a vector, we can figure out its angle for trigonometry.0472

So, once again, we have u = (3,4); so, tanθ = sideopposite/sideadjacent, when we have a right triangle.0477

And we have that here, because we know we are dealing with a nice rectangular coordinate system.0487

We have a 3 here and a 4 here; so our side opposite to our angle θ is this side here, and our side adjacent is this side here.0492

So, that gets us tanθ = 4/3; at this point, we can take the arctan of both sides0502

(the inverse tangent--arctan and tan-1 mean the same thing; I like arctan).0510

So, θ = arctan (tan-1) of 4/3; we plug that into a calculator or look it up,0514

and we see that that is approximately equal to 53.13 degrees; so we see that that is the angle in there.0520

Now, we normally talk about direction as the counterclockwise angle from the positive x-axis.0527

It is a little bit confusing, but it just means that we start over here at the positive x-axis,0532

and then we just keep turning counterclockwise until we get to whatever thing we are trying to measure out to,0536

at which point we stop, and that is the measure of the angle that we are going with.0542

It is just like we did with the unit circle: we started at the x-axis positive, and then we just kept spinning0546

until we got to whatever angle we were trying to get to.0552

But sometimes we won't be using the positive x-axis.0555

That is what we normally end up using, but sometimes the reference location--sometimes--will change,0558

and we won't be using this nice positive x-axis that we are used to using.0564

But maybe we will be talking about how far we are off of the vertical y-axis to the right,0567

or maybe some other thing, like we have this other angle here created,0574

and we are talking about how far we are off of this other thing, some angle, going clockwise...0579

So who knows how it is going to be done?0584

We have to pay attention to what the specific problem is and how it is set up.0585

It generally will be the positive x-axis, but it is not an absolute guarantee; you have to pay attention.0588

This means, in any case, whatever you are dealing with, that I recommend always drawing a picture,0593

because that is going to give you a way to be able to see what is going on.0599

Draw a picture before you try to figure out or use angles; it will help you get a sense of what is going on, and really help clarify things.0602

All right, now, what if we have the component from...0609

We talked about (previously) taking the magnitude and angle and getting component forms.0612

I'm sorry, we talked about the exact opposite of that.0619

So now, we are going to say, "What if we know the magnitude and angle, and we want to get the component form?"0621

So, if we are trying to find the component form from the magnitude and angle, we can figure that out.0626

The first thing first: always draw a sketch--it will help keep things clear and help you understand what is going on.0631

And then, from there, you just use trigonometry.0636

So, in this case, we have a length of 6; the length of our vector u is 6; our angle θ is 120 degrees.0638

We remember from trigonometry that the cosine of 120, the cosine of this angle here, is going to be equal to0647

the horizontal amount of change, up until it drops down on its crossover, divided by the length of the segment we are dealing with.0652

Cosine of 120 = x/6; multiply both sides by 6; 6cos120...cos120 is the same thing as -1/2, so 6 times -1/2...gets us -3.0662

So, -3 is our value for x.0673

A similar thing is going on for our y, the vertical component; sin(120), the sine of the angle we are dealing with,0676

is equal to the vertical component, divided by the length of the entire segment (6 in this case).0684

So, sin(120) = y/6; multiply both sides by 6; sin(120) is √3/2, so it simplifies to 3√3 = y.0689

At that point, we just take both of these pieces of information.0699

We slot them into our u, and we now have component form.0701

Our u is equal to (-3,3√3).0704

Now, I did tell us to do it with cosine 120; but most of us are probably used to dealing with trigonometry for degree angles under 90 degrees.0708

It is a little bit confusing--maybe just a bit--to be able to work with things over 90 degrees.0719

So, we also could have converted this to an angle of 120, so a total of 60 degrees over here, because it is 180 on the whole thing.0723

So, we could have figured out that inside of the triangle is 60 degrees.0732

Now, that is going to cause a little bit of difference here, because cos(120) brought that negative to the table,0735

because indeed, our x is going to the left, and remember: this is the negative direction for x.0740

Going this way gets us negative x values, when we go to the left, just like going down gets us negative y values.0745

If that is the case, we have cos(60), what we can figure out...0753

If you have things inside of a triangle, you are just going to figure out the length of each of those sides.0756

So, cosine of 60 equals x/6; so we will get 6 times...cos(60) is just 1/2, so we get that 3 = x.0760

But notice: what we are figuring out is...we are figuring out here to here as x, which is not the same thing as the x-value, as in the horizontal location.0772

What we are figuring out is just the length of the side.0780

We are figuring out how far it is from here to here; but we have to figure out the horizontal coordinates--not just the length, but the actual x-value.0783

If that is the case, what we are really figuring out is the absolute value of x--how long our x is.0792

And that means that, at the end, we have to look at this picture; and that is why we draw these sketches that are so handy.0798

We look at the picture, and we see that our x has a length of 3.0804

However, it is going to the left; so that means it has to be a negative thing.0809

By paying attention, we see that it is -3...not just length 3, but -3; but it is because we drew a sketch that we are able to see this.0816

You have this option: you can either just use the angle, and be able to be really good at trigonometry;0824

or you can make it in a slightly simpler form, where it is easier to understand.0828

But you have to be paying attention and realize that you have to set the sign at the end.0831

I have to pay attention, based on this sketch: is this length going to come out to be positive?0835

Is this length going to come out to be negative?0839

You really have to pay attention to that.0841

All right, we are ready to move on to a new idea.0843

Scaling by scalars: if we have a vector u, we can scale it to a different length by multiplying by a scalar, a real number.0845

A scalar is just some number; it is not a vector--it is just an actual, real number.0852

Algebraically, the scalar just ends up multiplying each component.0858

Let's say we start with this red vector on our picture, which is u = (3,-2).0862

We can scale this by some other thing, by just multiplying it by some number.0868

Like, for example: we could multiply u by 2, so we have 2; and now that is the blue vector that we see there.0872

Now notice that the blue vector is double the length of the red vector, because it is 2 times u.0877

So, it just takes that length, and it scales it by a factor of 2; it doubles that original length.0884

Algebraically, we just end up having this 2 multiply on the 3, and multiply on the -2; so we get (6,-4) as the components.0890

We can take this; we can try multiplying by something else--how about a negative number?0899

What would a negative get us? A negative ends up going in the opposite direction.0903

So, if we have a negative u, then the positive direction is the way it normally goes; negative will be the opposite way.0906

So, negative goes in the opposite direction; so now, we are going opposite the direction that u went, as we can see pictorially here.0913

And for how it is going to end up having it, it is just a negative now on each of the components.0919

So, negative cancels out there, and we have (-3,+2).0924

And that is what we see on our picture.0929

We could also have something that is not just a whole integer number, like, say, -3/2.0932

We end up having 3/2 times the length--1 and 1/2 the length of the original vector--0939

but then also, in the negative direction--opposite the direction of the first one.0946

Once again, numerically, it just ends up being -3/2 times the first number times the second number; so we get (-9/2,3).0950

That is what it ends up being.0959

So, algebraically it just multiplies each component; graphically it is a question of stretching and maybe also flipping.0960

In general, for any scalar k and some vector u that is (a,b), k times u...that is the same thing as k times (a,b),0966

so that is the same thing as just that k getting distributed to both the a and the b, so it gives (ka,kb).0976

Great; a unit vector is a vector with a length of 1; "unit vector" just means a length of 1.0983

It still has a direction; it can have any direction, but it has to have a magnitude of exactly 1.0990

Its magnitude--its length--how long it is...it is 1 in terms of length.0996

We create a unit vector out of any vector u by dividing u by its length.1000

Remember: if we divide by a number, that is the same thing as just using a scalar; we are multiplying by 1 over the number we are dividing by.1004

So, previously, we could scale; if we know its length is 10, and then we divide it by 10, we are going to get something with length 1 now.1011

We have some vector u that is some length; but then we divide it by that length, because the magnitude of u is just its length.1020

So, we have that; we divide by that; we have scaled it back to what it would be if it was just at a length of 1.1029

So, our unit vector is the original vector, divided by the length of the vector.1035

This is the same direction as u; so it will be in the same direction, but it is going to have a length of 1--it will just be length 1.1042

Now, the use of having a unit vector--the reason why it is so great to have a unit vector--1050

is that we can just take it later and multiply it by any scalar k that we want.1054

And we will know that we will have created a vector that is length k, because we started at length 1.1059

You scale that by k, and we are going to go with 1 times k; so we will just be at vector length k.1063

And we are going to be scaling in the direction we already started with, so we know that we will be in the unit vector's direction.1068

This gives us an ability to easily create vectors of any length we want in a known direction.1074

And this ability comes in really, really handy in a lot of situations, and that is why unit vectors are important.1079

All right, we can combine two or more vectors through addition or subtraction.1084

It is actually not that difficult; we just add them component-wise.1090

The horizontal components add together, and the vertical components add together.1093

All of your first components go together; the second components go together, and so on and so forth, like that.1096

In this case, if we add (3,4) and (2,-5), then what we end up doing is: we have the 3 and the 2 getting combined.1102

And they become 3 + 2, because we are adding (3,4) + (2,-5).1110

The same basic thing: the 4 and the -5 get combined, and so 4 + -5 is 4 - 5.1116

We simplify that, and we get (5,-1); we are just taking the numbers at the beginning and adding them together,1124

and the numbers at the end and adding them together.1130

We are doing it component-wise: each component stays and only mixes with components that are the same type1132

(first components with first components, second components with second components, and so on).1137

So, in general, for some (u1,u2)...really, that is just saying the first component of u,1142

and the second component of u...and then (v1,v2), the first component of v1147

and the second component of v--if we add u and v, u + v, (u1,u2) + (v1,v2),1153

it is going to be the exact same thing: the u1 and the v1, the first components,1158

adding together, and then the second components, u2 and v2, adding together, as well.1163

It is a very similar thing if we end up subtracting: u - v is just going to be u1,1172

the first component of u, minus the first component of v, v1;1177

u2 - v2 (the second component of u minus the second component of v).1180

The components stay in their locations, but they do either addition or subtraction, depending on whether it is addition or subtraction; it makes sense.1184

We can also do this geometrically and get an understanding of what is going on,1191

because vectors are supposed to represent this thing--this directed line segment.1194

So, we can see this idea geometrically: you add vectors by placing the tail of one at the head of the other.1198

So, for example, in this one, you have u in red, (3,4); and then we have v in blue, (2,-5).1205

So, we put (3,4) out to here; that goes out to (3,4).1212

And then, this one goes over 2 and goes down 5; so (2,-5) is 2 to the right, down by 5.1218

We put those together, and we get (5,-1), which is exactly what we see here in the purple vector that is the combination of them.1229

So, u + v is the purple vector that we see there.1236

We can see this algebraically and geometrically, and the two ideas match up completely.1239

Not only that, but we get the same resultant vector.1245

This resultant, what we get when we put the two things together, is the same whether it is u + v or v + u.1248

It doesn't matter if we end up doing the blue one first, and then the red one.1254

We can add them geometrically in any order we please, and it still comes out to be the same thing.1260

We end up seeing this parallelogram; we can see this as another way of adding things--creating this parallelogram out of them, or just head-to-tail.1264

But we see geometrically what is going on.1271

All right, now that we have these two ideas, we can put together the idea of combining vectors1275

with the idea of unit vectors to get a new way to express a vector's components.1280

We start by creating two standard unit vectors, which is really just a fancy way of saying "things that make sense and are kind of fundamental."1284

One horizontal, one vertical: i = (1,0); it is a unit vector that is purely horizontal;1292

and j, which is (0,1), is a unit vector that is purely vertical.1302

This one is just one unit long that way; and this one is just one unit long that way--purely vertical.1308

And that is what we are seeing there.1317

One other thing: if we want to write this bold thing, we can't really write bold on our paper; that is very difficult.1318

You can end up writing it like an i, but instead of putting a dot on it, you put that little arrow on top (or in my case, the harpoon on top).1324

The same thing with the j: you can make the j and then put a little arrow on top.1331

And so, that is another way of talking about these units vectors, i and j, if you want to.1334

All right, with this, we can express any vector in terms of i and j.1339

If we have (3,4), well, that is the same thing as having 3 i's, three of these unit vectors that are horizontal,1343

plus four of these unit vectors that are vertical; so that is what we have there.1350

We can combine them; we can break them up into 3 horizontal motions plus four vertical motions.1354

And we get to the same thing as if we had just done (3,4), all at once.1359

3, our first component, matches up with the i's, because that is just our way of saying our horizontal, since our first component is our horizontal.1363

And then, our 4 matches up with the 4j, because that is just the same thing as saying 4 vertical, or 4 units' worth of vertical.1371

And this also works for numbers that aren't just whole integers, as well,1379

because 4.7 times i just scales i by a factor of 4.7, so it will be in the same place horizontally.1381

And π--we can also scale by a factor of π; as long as it is a real number, we can scale by it; so πj goes in there; great.1390

All right, we can also talk about a zero vector; we denote the zero vector with that same arrow top on top of a 0.1397

That has 0 in all of its components; so it will be nothing but 0's as the vector.1404

It has no length; it is 0--it just lives at the origin.1408

And so, since it has no length, its direction doesn't matter.1412

So, here is an example: (0,0)...the zero vector is (0,0).1414

It is just sitting at the origin; its distance from the origin is nothing, because it is currently at the origin.1421

Notice: for any vector u whatsoever, if we add u and the zero vector together, we will just end up being there.1426

We will not have moved anywhere, because head-to-tail we end up going someplace,1432

and then we don't move, because the zero vector doesn't move at all.1436

u - u: if we subtract u from itself, we will end up going out and then coming right back, so we will end up getting 0.1439

And then finally, if we take any vector and multiply it by a scalar of 0, we will have some length,1446

and then we bring that length to 0; so at a length of 0, we have the zero vector; great.1450

All right, at this point, we have talked about a lot of different ideas with vectors.1455

And we can turn this into a bunch of properties.1457

Don't worry too much about understanding all of these properties right away.1459

They will just make more sense, and you will be used to using them.1463

The beauty of all of these properties is that they are very much what we are already used to using with the real numbers.1465

So, vectors, as long as you remember to keep them in this form of components only interacting with other components,1470

are very similar to working with numbers in many of the ways we are already used to.1475

Let's talk through some of these properties.1478

u + v is the same thing as v + u; this is the idea of commutativity, that 5 + 8 is the same thing as 8 + 5.1481

We are used to that with the real numbers.1488

We also have associativity, (u + v) + w is the same thing as u + (v + w).1490

3 + 5 + 4 is the same thing, if you add the 3 and the 5 first, or if you add the 5 and the 4 first.1496

3 + 5, then add 4, or 3 + (5 + 4) (where you add the 3 second)--it doesn't matter which way you do it;1502

so once again, it is very similar to doing it with normal numbers.1509

k times l...if we have two scalars, k and l, times u, well, that is the same thing as k times an already-scaled lu.1513

So, we can either multiply our scalars together and then multiply the vector;1520

or we can have them multiply the vector, each one after another--the same thing, either way, which is pretty much what we would expect.1523

k times (u + v) is nice; it distributes: k times vector u plus vector v is k times vector u, plus k times vector v.1530

It also distributes in the other way: k + l times vector u: the vector can distribute out onto them, so we have k times vector u, plus l times vector u; great.1539

If we take u, and we add it to the zero vector, we end up just getting u; it has no effect.1548

u - u is going to get us back to the zero vector.1553

And then, a couple more: 0 times u is going to give us the zero vector; 1 times u has no effect--1557

we are scaling by just what we are already at; and then, -1 times u is going to flip us to the negative version;1563

it will just cause everything in there to become negative.1568

The only one that might be a little bit confusing is ku, the magnitude of a scaled u, k scaled on u, k scalar times vector u.1570

The magnitude of that is equal to the absolute value of k, times the magnitude of u.1581

Let's look at why that is the case.1588

A really quick, simple example: let's consider if we had (0,1).1590

We have this vector here that goes from here out to a length of 1.1596

So, we could scale it by k; and let's say that k is equal to -2.1603

So, we scale this; here is our u: u = (0,1), so we could scale it by k = -2; -2 times u will get us the same thing,1609

but now it is going to be flipped, and it will be twice the length.1625

We are going to go down two units now.1629

So, whereas the first u went up by 1 unit (it had a length of 1), this has a length of 2.1631

The fact that we are going down doesn't make it negative length; length always is positive.1640

So, it has a length of 2, which is why we have the magnitude of (0,-2): well, that ends up being equal to +2, as we can see from this diagram right here.1644

But this -2 times u--if we had separated this out into -2 times (0,1), what u started as,1658

well, we could break this, by this rule, into the absolute value of our scalar, times the length of our initial thing,1672

which would give us positive 2 times 1--the same thing.1682

So, what it is doing is saying that the reason why we have absolute value on the scalar here1685

is because it doesn't matter that we are flipping and pointing in a new direction;1690

ultimately, length is always going to come out to be positive.1694

So, we can't let a negative k cause our result to come out as a negative length, because that just doesn't make sense.1697

So, we have to figure out a way for it to always stay positive, and that is why we have this absolute value here.1703

All right, there is no multiplication between vectors.1708

Even with everything we have seen so far about vectors, there has been no mention of vector multiplication, other than scalars multiplying on vectors.1712

But other than that, we haven't talked about vector multiplication.1720

That is because there is no good way to define vector multiplication.1722

There is just no way to really do it that is going to make sense.1727

So, we could make up some numerical way to multiply vectors: some vector times some vector makes some other vector.1730

But it would probably be geometrically meaningless; we can't really come up with a good way that is going to have some deep geometric meaning.1736

And that is the problem here: while we could come up with something numerically,1743

we want all of this stuff to have a geometric connection, and all of this other stuff has.1746

It makes sense to combine vectors; we are doing one vector, and then we are doing another vector.1750

We are doing two pieces of motion.1754

Or we stretch them: we have some piece of motion, and then we just elongate it or shrink it or flip it.1756

Those things make sense geometrically; but what would it mean geometrically--1760

what would it mean as a picture--to multiply a line segment by another line segment?1763

It just doesn't really make sense; and because of that, we do not define vector multiplication.1767

There is just no vector multiplication, pretty much, to talk about.1772

Now, all that said, there is an operation similar to multiplication that is called the dot product.1775

That is different, though, because it will take two vectors and multiply them together (although it won't multiply them together)--1781

it will take two vectors, and this vector dot, this other vector, will give us a scalar.1786

It will give us a single real number.1790

Don't worry about that too much now; we are not going to talk about it in this lesson.1792

But we will explore it in the next lesson; so we will see it soon.1795

But for right now, there is just no way to multiply vectors.1799

And even later on, once you see something that is kind of close to it,1802

you will see that it is very different from actually multiplying vectors and getting a new vector out of it.1805

All right, motion in a medium: a really common use of vectors is to analyze motion--1810

the location of an object, the velocity of an object, the acceleration of an object.1816

However, what happens if something is moving relative to a medium, like water or air?1820

Say we have a boat in a river, and so the boat is moving up the river.1826

But at the same time, the water in the river is moving in another direction.1830

We have to do something to take this into account.1833

So, the object is moving relative to the medium, but the medium itself is also moving, like the boat in the water.1835

So, to understand this, let's consider a fish swimming in an aquarium.1843

I love this as an example.1847

First, we have that the aquarium is completely still; we have some table like this;1849

and imagine that my arm here is the aquarium, and so here is the table.1854

The fish is here, and the fish is swimming forward; the fish swims forward, and it gets to the other end of the aquarium.1858

That is how it starts in this picture here: the fish is the only thing moving.1863

But what happens if we don't just let the fish be the only thing moving?1867

If we grab the aquarium, and we actually slide it to the slide as the fish is swimming--1871

if we grab the aquarium, and we move it, we are going to see the aquarium move like this while the fish is moving like this,1877

because the fish is moving inside of the aquarium, but now the entire aquarium is also moving.1884

We see that the fish moves, but at the same time, the aquarium moves over.1889

So, it is very different from the world where the fish ended over here.1893

Now, the fish manages to move at the same time as the aquarium is moved.1896

So, we have to take both of these things into account.1900

We can see this pictorially here: our fish is moving to the side, but at the same time, the box is moving to the side, as well.1902

The fish manages to get over to the left, but the box is now way over to the right.1911

So, if we are going to talk about where the fish has gotten to, the velocity of the fish--1916

anything where we want to talk about the fish's motion and analyze the motion of the fish--1919

we have to take both of these things into account.1924

We have to take into account the aquarium's motion, but also the fish moving inside of the aquarium.1926

It is not enough to talk about just one of them; we have to combine these two ideas.1931

Motion in a medium is the combination of the object's motion vector relative to the medium,1935

the fish relative to being inside of an aquarium, and then the medium's motion vector--how the aquarium is moving.1940

How does the fish move in the aquarium? How does the aquarium move in the larger world?1948

That is what we mean by relative to the medium; it is just whatever the thing is inside of,1952

and then how the thing that you are inside of is moving.1956

So, we can break this down into the velocity of the object, plus the velocity of the medium, is equal to the velocity of the total motion.1960

The fish relative to the table is the addition of the velocity of the fish in the water, plus the velocity of the aquarium,1967

as vectors, because the velocity vector in one case is going to be positive;1976

in the other...for one of them, it will be positive; for the other one, it will be negative,1980

because they are going to be pointing in opposite directions.1983

All right, we can also talk about more than two dimensions.1986

At this point, we have only seen vectors in two dimensions; but we can expand this idea to any number of dimensions we want.1990

For example, a three-dimensional vector could be (5,-2,3)--no problem.1995

We will just keep putting in more components.1999

By the way we define scalars and vector combinations, everything we have discussed so far about vectors still works just fine.2001

It might get a little confusing to picture in our head in higher dimensions.2007

But everything still makes sense; it is hard to picture higher than three dimensions,2010

because we are used to living in a three-dimensional world, but it still makes sense in terms of the algebra of what is going on.2013

That is great; also, one little thing: if we are talking about three dimensions...you remember that i and j--2019

we could talk about unit vectors as an alternate way of talking about component form.2025

There is another standard unit vector; there is k, which is (0,0,1).2028

So, i is the first component; j is the second component; k is the third component.2033

With this, we can express three-dimensional vectors with (i,j,k), so (5,-2,3) would become 5i - 2j + 3k.2037

5 becomes 5i; -2 becomes -2j; 3 becomes 3k; great.2046

So, we can combine them in terms of standard unit vectors, as well.2051

We can also talk about the magnitude of something that is higher than two dimensions.2055

It might seem a little surprising at first, but it turns out that it is actually really easy to figure out the magnitude-- the length--of a vector in any dimension.2058

Consider the n-dimensional vector x, where it is x1, the first component of x,2065

comma, x2, the second component of x, all the way up until we get to xn, the nth component of x.2072

Now, it turns out that the magnitude of our vector x is just the square root of the sums of each of its components, squared.2078

Now, that seems a little bit confusing, but it makes sense.2085

The length of our vector x is equal to the square root of the first component squared, plus the second component squared,2087

plus...all the way up until the nth component, our last component, squared.2093

So, that seems really surprising, the very first time we see this.2098

We will actually explore why this is the case, and it will make sense why this has to always be the case,2100

if we expand the idea that we will see in Example 3 of higher dimensions.2104

Just think, "Oh, yes, that will just keep stair-stepping up, and that is why we see this square root of all of the components, squared and added together."2108

But you can also just memorize this formula, if you want to.2114

And it will be just fine, too, and work out.2118

All right, let's see some examples.2120

First, given that u = (1,3), v = (4,2), w = (-5,1), what are each of the following?2122

(1,3)--if we are talking about u + v, then that is the same thing as talking about (1,3) + (4,2).2129

So, remember: we add the components together, so it will be 1 + 4, because they are both the first components;2138

and then 3 + 2, because they are both of the second components.2143

1 + 4 gets us 5; 3 + 2 gets us 5 also, just by chance; so our answer here is (5,5).2146

Next, we have 6u: well, 6 times...our vector is (1,3), so the 6 distributes effectively.2154

It is not quite distribution--it is a little bit different--but it has the exact same effect and feel.2162

So, the 6 multiplies each of the components; so we have 6, comma, 6 times 3 is 18; so (6,18) comes out of that.2167

All right, we can probably start doing these scalars in our head; they are not too hard to do.2175

1/2 times v...well, v was (4,2), so 1/2 times (4,2) is going to produce 2 (1/2 of 4 is 2), and 1/2 of 2 is 1.2179

So, we have (2,1) for 1/2v; and then, minus...our w was (-5,1).2189

We can distribute this negative here; so this becomes positive; this will become positive; this will become negative.2195

We distribute that negative into there; and so, 2 + 5 is 7; 1 - 1 is 0.2200

There we go; and the last one: if we have 2u - 3v + w...2208

All right, 2 times u: 2 times (1,3) will become (2,6); minus...3 times (4,2) will become (12,6), plus w: w was (-5,1); great.2211

(2,6) -...so that would make that -12 + -12, comma, -6, plus (-5,1).2229

At this point, we could add them all together; we could add them one by one; it doesn't really matter how we approach this.2243

Let's just add the first two; so 2 and -12 becomes -10; 6 and -6 becomes 0; plus...bring down the rest of it...(-5,1)...2248

So, (-5,1) + (-10,0): -10 and -5 becomes -15; 0 and 1 becomes positive 1; and there it is.2259

So, that is the basics of vector addition and multiplication by scalars.2269

It is very similar to what we are used to doing with numbers normally.2273

It is just that everything stays inside of its slot; they only interact with other things from the same slot as them.2276

First slots interact; second slots interact; and if it is higher than two dimensions, third, fourth, fifth...whatever slots interact.2282

All right, the next example: If u = 4, v = 6, and the two vectors make angles of θu = 30 degrees, θv = 120 degrees,2289

to the positive x-axis, what is the component form of u + v? its length? its angle?2297

All right, let's do u in red; over here, we first draw a sketch to be able to figure this out.2302

So notice: if we are going to figure out u + v, if we are going to get the component form of u + v,2309

well, it is hard to add angles and lengths together.2313

We could draw...well, u is an angle of 30 degrees, so we will go out like this.2316

And then, v is 6 at 120 degrees, so we will be a little bit longer...6...120 degrees...like that.2322

And so, our final thing will end up being this; and we could measure what that is.2329

But we would have to have really, really precise stuff; we would have to have a really accurate ruler,2333

and be doing this with a protractor that was really good, and be really, really careful to get all this.2338

So, it is not the sort of thing where we could draw it out and get a very good answer.2342

So, our first step is to get a component format of each of them, because once we have a component form for u2345

and a component form for v, it is easy to add them together.2350

And we add them together, and then we can figure out the length and the angle.2353

So, our first step is to get a component form.2355

u = 4; our θu is 30 degrees; and remember, it was with the positive x-axis.2358

So, our angle is 30 degrees like this.2363

So, if we want to break down u into its x-component and its y-component, well, then, we know that cos(30)...2369

remember, its hypotenuse was 4, so cos(30) is going to be equal to the x-component of u,2378

the first component of u (here is ux); the side adjacent...cos(30) = side adjacent, ux,2388

over the hypotenuse, 4; so 4 times cos(30) = ux.2396

cos(30) is just the same thing as √3/2, so we have 4(√3/2); so we have 2√3 = ux.2403

A very similar thing is going on if we want to figure out what uy is.2413

It will be sin(30) = uy/4; multiply by 4 on both sides; 4sin(30) = uy.2417

4sin(30)...sin(30) is just 1/2, so 4(1/2)...so we have 2 = uy.2428

So, at this point, we have that u equals 2√3, its x-component, comma, and its y-component, 2.2436

And that is what our values for u are.2448

Now, I want to point something out before we keep moving.2452

Notice right here: we were starting at sin(30) = uy/4, but we always get to this 4 times sin(30).2454

If you know the length of your vector, and you know what angle it is at, you can actually just hop to length of the thing,2463

times cosine (or sine, if it is side adjacent or side opposite, respectively) of the angle.2471

And that will just give you what that side adjacent or side opposite is, respectively.2478

So, we will end up doing that on the next one.2482

If it was a little bit confusing, notice the parallel to how we just did it with the u vector while we are working on the v vector.2484

But it is a really great way of being able to do this really, really quickly--2492

well, not really, really quickly, but it does help speed things up.2497

And it is a good trick, because you end up seeing this quite a lot.2500

So, we have 120 degrees here; we are at 120 degrees.2503

Now, I think it is going to be a little bit easier to figure out in terms of this angle here,2507

because it is easy to work under 90 degrees, so that is 60 degrees, because 120 + 60 = 180.2513

So, now we want to figure out the vx component, the horizontal component, and the vy, the vertical component.2521

So, vx is going to be equal to sine...it is 60...we have 60 in the angle,2529

and it is going to be side adjacent, so it is going to be the length, 6, times cosine, side adjacent of the angle involved.2536

However, there is one thing that we need to notice.2545

What we are figuring out here is the length of that side of a triangle, because the angle is inside of a triangle.2547

So, it is up to us to pay attention: is it going to be positive? is it going to be negative?2553

Normally, we have the x direction going negative as we go to the left.2556

That is negative x going this way, so that means we have to have a negative sign on our x-component for v.2561

Otherwise, it won't make sense, because we can see from the picture that it is going negative on the horizontal.2568

So, we have to pay attention to this; we have to notice this stuff happen, because otherwise it will be a mistake.2573

OK, -6 times cos(60)...cos(60) is 1/2, so we get -3; vx = -3; vy = 6.2577

This one is positive, because it is going up...sine of 60 is √3/2, so that gets us 3√3.2589

So, we have that the x-component is -3; the y-component is 3√3.2599

So, our v vector equals (-3,3√3).2606

Now, if we want to add these two together, u + v, it is simply a matter of adding them together.2613

Our u was (2√3,2); our v was (-3,3√3); we add them together; we have (2√3 - 3, 2 + 3√3).2620

And if we want to, we could get what that is approximately, in terms of a decimal number, although that is exact.2636

That is a perfect thing; this will just be approximate, because it is a decimal of square roots: (0.464,7.196).2641

All right, so at this point, we want to figure out what its length is.2651

The length of u + v is going to be equal to the square root of each of its components, squared and then added together.2654

So, its two components were 0.464, squared, plus 7.196, squared.2665

What does that end up coming out to be?2673

We work that out, and that ends up coming out to be...sorry, I couldn't find it in my notes...7.211.2676

It comes out to be approximately 7.211.2688

If we want to figure out what the angle is that it is at, the first thing we probably want to do is draw a quick diagram, so we can see it.2690

So remember: it was at this one right here, in terms of its components.2696

0.464 is just a little bit over the x-component, and then 7.196 up...so it is like this.2701

So, we can see that that should be what the angle is like.2707

So remember: tan(θ) is equal to the side opposite, divided by the side adjacent.2710

The side opposite in this case will be the vertical height, divided by the side adjacent of 0.464, the horizontal amount.2717

Take the arctan of both sides, the inverse tangent of 7.196 over 0464.2726

We plug that into our calculator or look it up in a table; and that comes out to be approximately 86.31 degrees.2736

Great; all right, the next one: what is the magnitude of the vector u = (4,3,12)?2744

And then, we want to figure it out by both the formula we were given and the Pythagorean theorem.2752

First, the formula we have--this is the easy part, a nice, handy formula.2757

It is going to be the square root of each of its components squared (4, 3, and 12), and all added together.2762

We work this out; we get √(16 + 9 + 144) =...add those together; 16 + 144 gets us 160; 160 + 9 gets us 169,2772

which simplifies to √169, is 13; so there is the length of our vector.2786

Now, we also want to figure out the Pythagorean theorem.2793

This is where we are going to understand why this formula--why this mystical formula actually works and makes sense all the time.2795

First, let's see where this vector would get plotted out to.2802

So, we are going to have to look at this three-dimensionally.2806

And while we haven't talked about three-dimensional coordinates before in this course,2807

you have probably seen them at some point previously in some course.2811

Here are our x, our y, and our z-coordinates.2813

We go out: (4,3,12); so 4 out on the x...a little way out on the x, a little less out on the y...2817

We are out here: 4, 3, 4, 3...and then we go up by 12; so our vector is like that.2826

Now, notice: we had to get out here to this place by the x and the y part first.2837

We could figure out what the length is here, and then we have a square angle here, as well.2843

So, let's figure out...we can break this down into two parts: what is happening in the (x,y) plane...2849

in the (x,y) plane we have (4,3) as the cross-section here.2855

I will color it: this part here is the same as this part here.2863

We figure that out; that is 3 there, as well; we are going to end up getting (by the Pythagorean theorem) √(32 + 42).2871

That equals √(9 + 16), equals √25, equals 5; great.2880

We have figured out what the lower part is on the bottom part.2889

Now, we can do this cross-section with the z-axis included.2892

So now, we look at a cross-section; using this cross-section, we can cut this, and we can see:2898

here is the thing we are trying to figure out, the length of this.2908

And we know that the z amount was 12; so a cross-section with the z-axis...this here maps to this part here.2912

And then, here, our purple part shows up here.2922

That was length 5, as we just figured out.2927

We use the Pythagorean theorem here; so once again, the value of our hypotenuse is going to be the square root of 122 + 52.2929

So, that is the square root of 144 + 25, or the square root of...add those together; we get 169, which equals 13.2938

That is the exact same thing that we saw over here when we used that formula.2948

Cool--it works out both ways.2952

Now, let's understand why it works out both ways.2953

Well, notice: the thing here for the purple line was the square root of 32 + 42.2956

It was the square root of the x2 plus the y2, the first component and second component put together, squared.2962

Now, notice: we end up just plugging in the purple part, because it makes up one of the parts of our triangle here.2969

We could have alternatively looked at the square root of 122, plus the purple part,2974

because that is what is going to go in there: the square root of 32 + 42,2983

because that is what the purple part ends up being;2988

and then, because we are going back to using the Pythagorean theorem, that whole thing is going to be squared, as well.2990

Well, that means √(122 plus...if we have the square root then being squared,2995

squared on top of square root, that cancels out, and we have + 32 + 42.3000

And look: that is the exact same thing that we have up here.3009

And so, that is where we are getting the ability to just put them all together--stack them all together.3012

It is because we have to figure out one cross-section after another after another.3016

But if we then plug in the way these cross-sections end up working out, each of the square roots3020

that would go in from a cross-section would get canceled by the next cross-section it goes into.3023

And so, ultimately, we end up getting this form of first component squared, plus second component squared,3027

plus third component squared, until we get to our last component squared.3032

And then we are adding them all and taking the square root, and that is why we have that formula for the magnitude.3035

All right, next: A box weighing 300 Newtons is hung up by two cables, A and B.3040

Using the diagram, figure out how much tension is in each cable.3045

The first thing to do is to understand what this means.3049

A lot of math problems get thrown at us like this, where they are actually pulling from physics.3052

And they are sort of assuming that we know things about physics that we might have no idea about.3056

This is a math course, not a physics course!3059

Let's first get an understanding of what this means.3061

If we have a box on a string--just imagine for a minute; now imagine that it weighs 100 Newtons.3065

And if you didn't know, a Newton is the unit of force and weight in the metric system.3072

They use kilograms for mass, but force, how hard something is being pushed or pulled--that is Newtons.3080

In the English system, the British imperial system, it is pounds; so pounds is used for weight and force,3085

although we actually have another unit for mass; but you almost never hear it.3092

It is called slugs, if you are using the British imperial system.3095

But Newtons are a way of measuring weight, which is just a question of how much gravity is pulling.3098

OK, so imagine that this thing is being pulled down by 100 Newtons of gravity.3104

We have 100 Newtons of force pulling down on this thing.3110

Well, if we have some rope or some string that is holding this thing up,3112

well, if it is being pulled down, it must be that the string is pulling back up.3116

Otherwise, the thing would fall to the ground.3120

So, how much is the string pulling up by?3123

Well, the string must have a tension pulling up of 100 Newtons in the opposite direction; so there are 100 Newtons going up and 100 Newtons going down.3124

Now, notice: as a vector, this would be a positive 100 Newtons, because it is in the up direction.3132

As a vector, this would be a negative 100 Newtons, in the down direction.3138

We take positive 100 Newtons, and we add that to -100 Newtons; we get 0.3142

This makes sense, because no force means no acceleration.3148

The thing is currently stopped; so as long as it doesn't have any acceleration to make it move somewhere, it is going to not pick up any motion.3155

What we have to have: we have to have no force out of it for it to not move.3161

Now, since it is hung up by two cables, it is perfectly reasonable to say, "Yes, if it is hung up, it is not currently moving anywhere."3165

It is not falling to the ground; it is not swinging left and right; it is just hanging there in space--it is sitting there.3171

So, that means that it must be a total of 0 for what is going down and what is going up.3175

We know that it is 300 Newtons going down; so now it is a question of how much is going up.3180

So now, we need to talk about these cables.3184

We can think about A as being a vector pulling out and away, because it is pulling up on that box.3187

Otherwise, it would be helping the box fall.3194

So, this is some vector A, and it has some force tension; so we will say A is equal to the tension in A.3196

And we will do the same thing over here with B; so now we have some vector B,3206

and B will be the amount of the tension in B, how much it is being pulled up by.3211

Now, we don't know what A and B are; that is what we are trying to figure out.3217

But we want to figure out how to get to them; so we start looking at this, and we say, "Well, I don't know a lot."3221

But they did tell us this information about the angles.3227

So, maybe we can break these vectors up into component forms, based on these angles.3230

So, if we have 40 degrees here, then we must have 90 - 40 here.3236

And since this is another right angle, that means we have 40 here again.3244

Similarly, with the same basic idea, since it was 70 up here, then we have 70 down here.3248

With that in mind, we can figure out what the pieces are here.3255

We know that A is the length of this hypotenuse; we don't know what the number is yet, but we are calling it A.3261

So, the vector A is going to be broken into the components: the horizontal amount is the side adjacent to 40,3267

so A, the length of the whole thing, times side adjacent of the angle;3273

and then, A times opposite, if we want to talk about the vertical part right here.3279

That will be Asin(40) over here.3285

Next, we know that we can talk about vector B; we can break it down into very much the same way.3289

That will be B times cos(70), because its side adjacent is 70; and B times its side opposite...3295

I'm sorry, not its side adjacent; its side adjacent is not 70, but the angle connected to side adjacent is 70 degrees.3304

And B times sin(70)...we can get this all just from basic trigonometry stuff.3308

Here is B; we can multiply B, our hypotenuse, and figure out what the sides opposite are, based on this.3314

We have two vectors here, A and B.3320

One other thing that we know is the force vector: what is the force on this box?3323

Well, it must be that the force on this box...is it moving up and down? Is it currently still?3329

Well, it is currently still; it is not moving anywhere; and it is not moving anywhere horizontally; and it is not moving anywhere vertically.3333

So, total force, in the end, once everything gets put together: it is not moving up and down; it is not moving left to right; so it must be (0,0).3342

And they did tell us one other piece of information, 300 Newtons; so the weight is equal to 03349

(because it is perfectly down, so it doesn't have any horizontal) and -300, because it is moving down; great.3357

At this point, we actually have enough information to solve it; let's work this thing out now.3362

Remember: we have weight = (0,-300); and we know that when we combine the weight with each of the cables--3367

once we put all of these forces together, all of the forces put together must come out to be a force of nothing,3378

because the box isn't moving anywhere--it is not being shoved around.3384

We can plug all of these things together; we have A + B + the weight vector is going to equal our total force of (0,0).3387

So, let's work out what that is: we have -A...3408

Oh, that was one thing I didn't say on the previous slide; I put it as Acos(40), because Acos(40) is the length of this side.3411

But it has to be negative, because remember: when we go to the left, it is negative in the x direction, so it is negative.3418

Both of the verticals are positive, because they are both pointed up; but only the B is pointed to the right in this--pointed positively horizontally, as well.3425

All right, we have (-Acos(40), Asin(40)), plus (Bcos(70),Bsin(70)).3433

I'm sorry; I am going to have to continue onto the next line.3449

Remember: this is just all one line put together...so + (0,-300).3452

In the end, we will end up equaling (0,0).3457

Let's combine this all together; we will switch to the color of green for everything together.3462

We have -Acos(40) + Bcos(70) + the 0 from the weight; that comes out to be the total for our first components.3467

And then, Asin(40) + Bsin(70) - 300 = (0,0).3480

So, at this point, we know that the first component on the left side of an equation3492

has to be the same as the first component on the right side of the equation.3496

The same thing: the second component on the left side must be the same thing as the second component on the right side.3499

So, we can break this down into two separate equations.3503

We have -Acos(40) + Bcos(70) = 0; so we can get B on its own or A on its own, and then plug that into the other one.3505

Let's solve for that first: Bcos(70) = Acos(40); at this point, we see that B is equal to A times cos(40), over cos(70).3519

We could plug that into a calculator right now and get some number out of it.3532

But then we would have to write all of these decimals; so we can just leave it like that for now.3535

Next, we will swap to a new color for solving this part.3538

We have Asin(40) + Bsin(70) - 300 = what was on the right-hand side, 0.3542

So, we can move the 300 over; we see that we have Asin(40) + Bsin(70) = 300.3554

Now, we see that B is the same thing as A times cos(40), divided by cos(70); so we have Asin(40) +...3564

we swap out our B: (Acos(40))/cos(70), times sin(70), still equals 300.3572

So, at this point, we pull out all of our A's; we have A times sin(40), plus cos(40)/cos(70), times sin(70), equals 300.3585

Notice that there is nothing we can cancel out there, because we don't have any exactly matching things.3599

But we can divide by it: 300 divided by sin(40) plus cos(40) over cos(70) times sin(70).3604

We can plug that all into a calculator, and it will come up and give us an answer.3622

And it will tell us that A is approximately equal to 36.40...oops, sorry, not 36.40, but times 3...3626

give me just a second...the magic of video...it comes out to be approximately 109.2.3640

And that is our value for A; to figure out our value for B, we have this handy thing right here.3648

B equals A, which was 109.2, times cos(40), divided by cos(70).3654

We work that out with our calculator, and we get approximately 244.6.3663

So, the tension in B is 244.6 Newtons, and the tension in A is 109.2 Newtons; great--those are our solutions.3670

All right, the last example: A plane has a compass heading of 75 degrees east of due north, and an airspeed of 140 miles per hour.3689

If the wind is blowing at 20 miles per hour, and towards 10 degrees west of due north, what is the plane's direction and speed, relative to the ground?3697

First, what does this first part mean?3704

We have a compass heading of 75 degrees east, going at 140 miles per hour airspeed.3705

Airspeed means your speed in the air--how fast you are moving, relative to the air right around you.3711

So, if that is the case, then our plane is moving 75 degrees east of due north.3716

Due north is this way, our vertical axis; so if that is the case, we need to curve 75 degrees down towards the east.3726

So, east is this way, so that is 75 degrees here.3735

If we want to figure out what is the thing in here, because we will probably want that for figuring out other things,3739

that is going to be 15 degrees; so it is 15 degrees for our normal θ that we are used to.3743

OK, so that is the direction that the plane is headed; it is going like that.3750

It is going off in this way; but then, that is its airspeed.3754

The air also is able to move; the air is going this way--the air is blowing the plane.3758

So, the plane is going like this, but at the same time it is being blown off-course slightly by the wind.3763

Or perhaps (hopefully) they have taken this into account, and it is not going off-course.3768

We have the wind blowing at 20 miles per hour, and towards 10 degrees west of due north.3772

So, once again, it is off of due north; and now it is just a little off, 10 degrees off, here.3778

And it is total of 20 miles per hour.3785

So, we have 140 miles per hour this way and 20 miles per hour, 10 degrees to the west of due north.3787

So, notice that the total angle for that is going to end up being 100 degrees, if we wanted to figure that out.3795

Or we could also look at it in terms of 80 degrees here, as well.3802

It depends on which one you think is easier; I am going to go with the one inside of the triangle,3809

and we will just have to remember to deal with the fact that our horizontal is going to be negative when we are working on the wind.3813

So, the plane--what is the velocity of the plane in the air?3818

To be able to figure out what the plane's direction and speed is relative to the ground,3823

we have to combine its motion in the medium with the medium's motion relative to the ground.3828

Its motion in the medium is the 75 degrees east of due north and an airspeed of 140 miles per hour, the blue part.3834

And the red part, the wind blowing, is the air relative to the ground; so we have to combine those two things.3840

So, the velocity of the plane in the air...its horizontal component is going to be the length of the vector, 140, times...3846

the angle we have is 15, so if we are talking about the horizontal, that is going to be side adjacent.3855

So, cos(15)...140 times sin(15) for the vertical, because that is side opposite...we work that out;3858

and that ends up coming out to be approximately 135.2 and 36.23.3867

And the units on both of those are miles per hour, because it is moving 135.2 miles per hour north and 36.23 miles east simultaneously.3875

And then, we have the wind: what is the velocity of the air itself?3884

The air is moving at a speed of 20; and if we want to figure out its horizontal component, it is going to be this part here,3889

at which point we say, "Oh, right; it is going negative, so let's put in that negative sign, because it is moving to the left."3896

We have to remember to catch those negatives.3903

Pay attention to if it is going to be a positive or a negative direction for everything.3906

So, that is going to be 20, the size of it, times cosine of 80 degrees, in this case;3909

and then positive 20, because this one is going positively, times sine of 80.3916

Work that one out with a calculator; and we get 131.7, 55.93.3923

Oops, I'm sorry--I wrote the entirely wrong thing.3932

I meant to write -3.47 (I read the wrong thing off of my notes), comma, 19.70; great.3934

All right, so if we want to figure out the combination of the two--if we want to figure out what the total motion of the plane,3944

relative to the ground, is, that is going to be the plane's motion relative to the air, plus the air's motion relative to the ground.3950

We just figured out what each one of those is: (135.2,36.23) was our plane's motion,3960

plus (-3.47,19.70); so in total, that gets us (131.5,55.93); great.3968

So, that is what the velocity vector is going to be--what the component form is.3981

However, it asked for speed and direction of the whole thing put together.3988

If that is the case, we want to figure out speed.3994

Well, speed is just the size of our velocity total vector.3997

That is going to be the square root of the first component of our total vector, 131.7, squared,4004

plus the second component, squared (55.93 squared).4009

We take the square root of that and figure it all out, using our calculator; and we end up getting 143.1 miles per hour.4013

The plane is actually going a little bit faster than its speed had been previously without the air connected to it.4021

However, its direction will also change.4028

To help us figure out direction, let's draw just a quick picture, so that we know what is going on.4030

We have this right here as our motion: 131.7 is fairly horizontal, and a little bit...between 1/3 and 1/2 of our amount horizontally up.4035

That is what our motion is like total.4048

My picture is totally not relative; 140 up here should be much longer than the 20 here;4052

and this 143.1 miles-per-hour long vector should be even longer.4057

But that is OK; we are just trying to get a sense of what is going on; they are sketches, not perfect drawings.4060

So, we are looking for this angle here: tan(θ) is going to be the side opposite, the vertical component, 55.93,4065

divided by the side adjacent, the horizontal component, 131.7.4075

We take the arctan of that; that gets us θ =...about 23 degrees.4080

Now, notice: that is what θ is equal to: θ equals 23 degrees.4085

All of this stuff was given in east of due north, so we have to put it in that same thing.4090

If it is 23 degrees as our θ here, that means it is 23 degrees going up from our positive x-axis, which was east.4095

Up would be towards the north; so we could phrase this as 23 degrees north of east.4102

Alternatively, if we wanted to use the exact same thing that they had done, when they all talked about east of due north,4112

90 - 23...if we want to figure out what this is here, 90 - 23 = 67.4119

So, we could also talk about it as 67 degrees east of north.4127

Either one would be fine; but we have to put it in the same format, because they didn't give us a θ previously.4138

We don't know where θ is based off of, so we have to make sure4143

that we are following this same pattern of east of due north, west of south, something like that.4145

We have to go in that same pattern.4150

All right, vectors are really, really useful; we will talk about them more when we talk about how matrices are connected to them.4152

But this is really great stuff here.4156

We have talked about a whole lot of things here, so if you had any difficulty understanding this lesson,4157

just try watching piece-by-piece, and just work examples, one after another.4162

There are a lot of things to digest here, but they all work together; and vectors are so useful.4165

All right, we will see you at Educator.com later--goodbye!4169