  Vincent Selhorst-Jones

Dot Product & Cross Product

Slide Duration:

Section 1: 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
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
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
18:50
19:04
19:21
20:04
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
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
Section 2: 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
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
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
Section 3: 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
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
13:08
13:21
Form of Factored Binomials
13:38
Factoring Examples
14:40
16:58
Factoring Higher Degree Polynomials
18:19
Factoring a Cubic
18:32
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
9:56
11:38
Derivation
11:43
Final Form
12:23
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
22:00
Example 3: Solve for Zeroes
25:28
Example 4: Using the Quadratic Formula
30:52

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
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
26:49
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
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
19:24
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
6:29
Non-Distinct Roots
6:59
Denoting Multiplicity
7:20
7:41
8:55
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
Section 4: 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
8:25
9:26
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
Section 5: 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
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
Section 6: Trigonometric Functions
Angles

39m 5s

Intro
0:00
Degrees
0:22
Circle is 360 Degrees
0:48
Splitting a Circle
1:13
2:08
2:31
2:52
Half-Circle and Right Angle
4:00
6:24
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
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
9:43
Mnemonic: All Students Take Calculus
10:13
13:11
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
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
Section 7: 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

52m 52s

Intro
0:00
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
Section 8: 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
Section 9: 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
11:42
Example of Using a Point
12:41
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
11:14
Example 1
13:24
Example 2
15:50
Example 3
22:02
Example 4
29:06
Example 4, cont.
33:40
Section 10: 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
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
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
Section 11: 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
Section 12: 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
Section 13: 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
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
Section 14: 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
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
11:51
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
Section 15: 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
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
Section 16: 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
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
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
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
Section 17: Appendix: Graphing Calculators

10m 41s

Intro
0:00
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
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
5:44
Which Graphing Calculator is Best?
5:46
Too Many Factors
5:54
6:12
The Old Standby
7:10
TI-83 (Plus)
7:16
TI-84 (Plus)
7:18
9:17
9:19
9:35
10:09
Graphing Calculator Basics

10m 51s

Intro
0:00
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
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
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
5:10
Find Out What Input Value Causes a Certain Output
5:12
For Example
5:24
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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• ## Related Books 1 answer Last reply by: Professor Selhorst-JonesSun Sep 18, 2016 1:00 PMPost by Khanh Nguyen on September 17, 2016I love the bonus round. 1 answer Last reply by: Professor Selhorst-JonesTue Jan 6, 2015 12:20 PMPost by Jamal Tischler on December 29, 2014I thought the cross product was UxV=u*v*sin(thetha) and than we use the right-hand-rule to see it's direction.

### Dot Product & Cross Product

• Given two n-dimensional vectors u = 〈u1, u2,…,un〉 and v = 〈v1, v2, …, vn 〉, the dot product (symbolized by a · between the vectors) is the sum of the products of each pair of components.
 →u · →v
 =
 〈u1, u2,…,un〉·〈v1, v2, …, vn 〉
 = u1v1 + u2v2 + …+ un vn
Notice the dot product results in a scalar, not a vector. [We say u ·v as  u dot v' when speaking.]
• The dot product of two vectors is deeply related to the angle between the two vectors (if put tail to tail) and their magnitudes.
 →u · →v =  | →u | | →v | cosθ
• We can interpret the dot product geometrically as projecting one vector on to the other, then multiplying the length of the resulting projection by the length of the vector being projected onto. In general, we can interpret the dot product as a measure of how long and how parallel two vectors are.
• From the formula above, we see
 →u · →v =0     ⇔ →u is perpendicular to →v .
[There are many equivalent words for perpendicular': normal', orthogonal', at right angles', but these all mean the same thing: θ = 90°. (It's a very important idea in math, so that's why there are so many synonyms.)]
• Unlike the dot product, the cross product only works in three dimensions. It takes two vectors and produces a third vector that is perpendicular to both. If u = 〈u1, u2, u3〉 and v = 〈v1, v2, v3〉, the cross product is
 →u × →v =  〈 u2 v3 − u3 v2,    u3 v1 −u1v3,    u1 v2 − u2 v1 〉.
[We say u ×v as u cross v' when speaking.]
• While we haven't learned about matrices or their determinants (yet!), you may have learned about them in a previous math course. A great mnemonic for remembering the cross product is with the determinant of a 3×3 matrix and standard unit vectors:
 →u × →v
 =
 i
 j
 k
 u1
 u2
 u3
 v1
 v2
 v3
 ←unit vectors
 ←first vector
 ←second vector
• Given two vectors u and v, the cross product produces a third vector u ×v that is perpendicular to both. However, notice that there are two possible directions such a perpendicular vector could point in. We orient the cross product based on the right-hand rule: point the fingers of your right hand in the direction of the first vector, with your palm in the direction of the second. The cross product's direction is the direction your thumb points.
• The magnitude (length) of u ×v is equal to the area of the parallelogram enclosed by u and v. In general, we can interpret the length of u ×v as a measure of how long and how perpendicular the two vectors are.

### Dot Product & Cross Product

Let u = 〈7,  12 〉 and v = 〈−3,  5 〉. What is u ·v?
• u ·v is the dot product of u and v. The dot product of two vectors is the sum of the products of each pair of components.

•  →u · →v =     〈7,  12 〉·〈−3,  5 〉    =     7 ·(−3)   +   12 ·5     =     −21 + 60     =     39
u ·v = 39
Let a = 〈−5,  8 〉 and b = 〈−7,  −11 〉. What is a ·b?
• a ·b is the dot product of a and b. The dot product of two vectors is the sum of the products of each pair of components.

•  →a · →b =     〈−5,  8 〉·〈−7,  −11 〉    =     (−5) ·(−7)   +   8 ·(−11)     =     35 −88     =     −53
a ·b = −53
Let u = 〈3,  9,  −5 〉 and v = 〈6,  0,  4 〉. What is u ·v?
• u ·v is the dot product of u and v. The dot product of two vectors is the sum of the products of each pair of components. It doesn't matter how many components the vectors have, as long as they have the same number of components. Just take each pair of components, multiply them together, then sum it all up.

•  →u · →v =     〈3,  9,  −5 〉·〈6,  0,  4 〉    =     3 ·6  +  9 ·0  +  (−5) ·4
We take each pair of components, then multiply them together. From there, it's just a matter of simplifying:
 3 ·6  +  9 ·0  +  (−5) ·4     =     18 + 0 −20     =     −2
u ·v = −2
Let u = 〈4,  8 〉 and v = 〈6,  −1 〉. What is the angle θ between u and v?
• The dot product of two vectors is connected to the angle between those vectors and their magnitudes. The following formula is very useful, and what we will need for this problem:
 →u · →v =  | →u | | →v | cosθ
For this problem specifically, we're focused on finding the angle between the vectors, so we can formulate the above as
cosθ =  →u · →v

 | →u | | →v |
• Therefore, to solve for θ, we need to know what |u| and |v| are. Let's find those:
 | →u |     = √ 42 + 82 = √ 80 =     4 √5

 | →v |     = √ 62 + (−1)2 = √ 37
• Plugging in to the formula:
cosθ =  →u · →v

 | →u | | →v |
=     cosθ =〈4,  8 〉·〈6,  −1 〉

 4 √5 · √ 37
Work out the dot product and simplify:
cosθ    =     4 ·6 + 8 ·(−1)

 4 √ 185
=     16

 4 √ 185
=     4

 √ 185
Finally, solve for θ:
cosθ = 4

 √ 185
⇒     θ = cos−1

4

 √ 185

Using a calculator, we get θ = 72.897°.
θ = 72.897°
Let a = 〈5,  −4 〉 and b = 〈8,  k 〉. What value must k be for a and b to be perpendicular?
• If a and b are perpendicular, then we know the angle between them must be θ = 90°. From the dot product, we know for any two vectors
 →u · →v =  | →u | | →v | cosθ.
Therefore, if θ = 90°, since cos(90°) = 0, it must be that the dot product of a and b comes out to 0:
 →a · →b = 0
• Since we know a ·b = 0, we can set up the following:
 〈5,  −4 〉·〈8,  k 〉 = 0
• From there, we can do the dot product
 5·8 + (−4)·k = 0,
then work to solve for k:
 40 −4k = 0     ⇒     4k = 40     ⇒     k = 10
k=10
Let u = 〈3,  −7,  −8 〉 and v = 〈2,  −5,  10 〉. What is the angle θ between u and v?
• Previously, we talked about an equation that connects the dot product of two vectors with the angle between them and the lengths of the vectors. This equation is true no matter how many components the vector has. While we're used to working with two-dimensional vectors, it makes no difference that u and v are three-dimensional this time: the equation is still just as true.
 →u · →v =  | →u | | →v | cosθ
For this problem specifically, we're focused on finding the angle between the vectors, so we can formulate the above as
cosθ =  →u · →v

 | →u | | →v |
• Therefore, to solve for θ, we need to know what |u| and |v| are. Let's find those:
 | →u |     = √ 32 +(−7)2 + (−8)2 = √ 122

 | →v |     = √ 22 + (−5)2 + 102 = √ 129
• Plugging in to the formula:
cosθ =  →u · →v

 | →u | | →v |
=     cosθ =〈3,  −7,  −8 〉·〈2,  −5,  10 〉

 √ 122 · √ 129
Work out the dot product and simplify:
cosθ    =     3·2 + (−7)·(−5) + (−8) ·10

 √ 122 · √ 129
=    −39

 √ 122 · √ 129
Finally, solve for θ:
cosθ =−39

 √ 122 · √ 129
⇒     θ = cos−1

−39

 √ 122 · √ 129

Using a calculator, we get θ = 108.112°.
θ = 108.112°
Let p = 〈4,  0,  −3 〉 and q = 〈−2, 5,  7 〉. Find p ×q. Verify (by the dot product) that p ×q is actually perpendicular to p and q.
• The cross product of two vectors produces a new vector that is perpendicular to the two vectors used in the cross product. It only works in three dimensions, and is formulated as below:
 〈u1, u2, u3〉·〈v1, v2, v3〉 = 〈u2 v3 − u3 v2,   u3 v1 −u1v3,   u1 v2 − u2 v1 〉
• This means we need to use the components of p and q based on their locations within the vector-whether they are first, second, or third. Based on the above formula for the cross product:
 →p × →q =     〈0 ·7 − (−3)·5,    (−3)·(−2) − 4·7,    4 ·5 − 0 ·(−2) 〉
• Simplify:
 →p × →q =     〈15,   −22,   20 〉
• To verify that p ×q is actually perpendicular to p and q, we can use the dot product. Remember, if the dot product of two vectors is 0, then those two vectors must be perpendicular. Therefore we can check for perpendicularity by making sure the dot product indeed comes out to be 0.

(p ×q) ·p:
 〈15,  −22,  20 〉·〈4,  0,  −3 〉    =     15 ·4 + (−22) ·0 + 20 ·(−3)     =     0

(p ×q) ·q:
 〈15,  −22,  20 〉·〈−2, 5,  7 〉    =     15 ·(−2) + (−22) ·5 + 20 ·7     =     0
p ×q = 〈15,  −22,  20 〉 To verify that p ×q is perpendicular to p and q, compute the dot products with each and show that they are equal to 0.
Find the area enclosed by the parallelogram where the vectors 〈2,  7〉 and 〈−8,  1〉 make up two of the sides. • We could approach this problem by finding a formula that gives the area of a parallelogram based on some of its measurements, then use a combination of trig and geometry to find those elements. However, using what we know about vectors, there's an easier way! For any two vectors u and v, the magnitude of their cross product u ×v is the area of the parallelogram they enclose.
• First, we check how to find the cross product:
 〈u1, u2, u3〉·〈v1, v2, v3〉 = 〈u2 v3 − u3 v2,   u3 v1 −u1v3,   u1 v2 − u2 v1 〉
Ack! There's a problem! The above uses three-dimensional vectors, but the vectors we are working with are only two-dimensional! Luckily, there's an easy fix: we can give them an extra dimension, but just say they have 0 in that component. After all, in a way, everything in two-dimensional space is inside a three-dimensional space: it just doesn't poke off the plane into that third dimension. Thus, we can use the below vectors:
 〈2,  7〉    ⇒     〈2,  7,  0〉 ⎢⎢ 〈−8,  1〉    ⇒     〈−8,  1,  0〉
• Now find the cross product for those vectors:
 〈2,  7,  0〉×〈−8,  1,  0〉    =     〈7 ·0 − 0 ·1,     0 ·(−8) − 2 ·0,     2 ·1 − 7 ·(−8)
Simplify:
 〈2,  7,  0〉×〈−8,  1,  0〉    =     〈0,  0,  58 〉
Then take the magnitude of the cross product:
 √ 02 + 02 + 582 =     58
The area of the parallelogram is 58 square units.
In physics, the work done by a force F over a distance d is defined as W = F ·d. You push a crate along the floor for a distance of 15 m. Your push has a total force of 200 N (newtons) and is directed at an angle of 20° below horizontal. What is the total work you have done on the box by the end? [The unit for work/energy in the metric system is the joule→ J.] • Since work is defined as W = F ·d, we need to find F and d.
• Finding d is quite simple. We know the box is pushed along the floor for a distance of 15 m. It doesn't move off the floor, so its distance vector is entirely horizontal:
 →d = 〈15,  0 〉
• Finding F is a little more complicated, but we can find it using some simple trigonometry. The length of its horizontal component (x) can be found as
 cos(20°) = x 200 ⇒     200 cos(20°) = x
Using a calculator, we get x = 187.94. We find the length of the vertical component (y) similarly:
 sin(20°) = y 200 ⇒     200 sin(20°) = y
Using a calculator, we get y=68.40. Now we can put these together to find F. However!, it is important to note that we only found the lengths of the components. We have to pay attention to the diagram to figure out their signs (+ or −). The box is being pushed to the right, so its x component will be positive. However, we see that the box is being pushed down (although it's against the floor, so it won't actually move down), so its y component is negative. Thus,
 →F = 〈187.94,   −68.40 〉
• Now that we know the vectors in component form for F and d, we only have to take the dot product to find the work:
 W = 〈187.94,   −68.40 〉·〈15,  0 〉    =     187.94 ·15 + (−68.40) ·0     =     2819.1
2819.1 j
Taking a nice stroll in the middle of a snowy, winter landscape, you see two children coming down a hill on a sled... but something looks wrong. Oh no! The sled is out of control! It's coming down way too fast, and the children don't know what to do! You are currently standing on a flat section of ground, and you see that the sled will come by near to where you are. The flat stretch of ground leads right to a clump of trees, and the sled will soon slam into them if you don't do something. Ever the hero, you decide to act. As the sled approaches, you notice a coil of rope on the back of the sled. At the moment the sled passes you, you snatch up the rope, and run along behind the sled, pulling backwards on the rope to provide some braking force on the sled. You pull backwards on the rope with a force of 140 N, and the rope makes an angle of 30° with the ground. You finally manage to bring the sled to a stop after it has slid a further 11 m. Much farther and it would have hit the trees, but thanks to your intervention, the children are safe! Congratulations! You're a hero! How much work did you do on the sled as it was brought to a stop? [Remember, work is defined as W = F ·d and the metric unit of work/energy is the joule→ J.] • First off-good on you! You really did an amazing thing, saving those kids like that. What a selfless act! Alright, now let's get down to analyzing what happened. To figure out the work involved, we can use the definition of work:
 W = →F · →d
Thus, to find the work, we need to know what the force (F) and distance (d) are in component form.
• Finding d is pretty simple. We know the sled moved a further 11 m after you began pulling back on it. It didn't move into the air, so it only moved horizontally. Thus, we have
 →d = 〈11,  0 〉.
• Finding F is a little more complicated, but we can find it using some simple trigonometry. We know the magnitude of the force is 140 N and the angle is θ = 30° because the rope is at that angle. Using that, the length of its horizontal component (x) can be found as
 cos(30°) = x 140 ⇒     140 cos(30°) = x
Using a calculator, we get x = 121.24. We find the length of the vertical component (y) similarly:
 sin(30°) = y 140 ⇒     140 sin(30°) = y
Using a calculator, we get y=70. Now we can put these together to find F. However!, it is important to note that we only found the lengths of the components. We have to pay attention to the diagram to figure out their signs (+ or −). You are pulling backwards on the sled, so because the horizontal goes to the left, the x component must be negative. You pull up on the sled, so the y component is positive
 →F = 〈−121.24,   70 〉
• Now that we know the vectors in component form for F and d, we only have to take the dot product to find the work:
 W = 〈−121.24,  70 〉·〈11,  0 〉    =     (−121.24) ·11 + 70 ·0     =     −1333.64
−1333.64 j [If you're confused why the answer comes out to be negative, pay attention to the direction of the force compared to the direction of the sled's motion. Also, it makes sense that the work done on the sled is negative: you took energy out of the sled to slow it to a stop.]

*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.

### Dot Product & Cross Product

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 0:08
• Dot Product - Definition 0:42
• Dot Product Results in a Scalar, Not a Vector
• Example in Two Dimensions
• Angle and the Dot Product 2:58
• The Dot Product of Two Vectors is Deeply Related to the Angle Between the Two Vectors
• Proof of Dot Product Formula 4:14
• Won't Directly Help Us Better Understand Vectors
• Dot Product - Geometric Interpretation 4:58
• We Can Interpret the Dot Product as a Measure of How Long and How Parallel Two Vectors Are
• Dot Product - Perpendicular Vectors 8:24
• If the Dot Product of Two Vectors is 0, We Know They are Perpendicular to Each Other
• Cross Product - Definition 11:08
• Cross Product Only Works in Three Dimensions
• Cross Product - A Mnemonic 12:16
• The Determinant of a 3 x 3 Matrix and Standard Unit Vectors
• Cross Product - Geometric Interpretations 14:30
• The Right-Hand Rule
• Cross Product - Geometric Interpretations Cont.
• 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.

### Transcription: Dot Product & Cross Product

Hi--welcome back to Educator.com.0000

Today, we are going to talk about the dot product and the cross product.0002

Now, we have some idea of vectors learned, so we can move on to looking at two new ways that vectors can interact: the dot product and the cross product.0005

These ideas are very important and useful in advanced math, science (especially in physics), and engineering.0014

If you have any interest in those fields, you definitely want to pay extra attention here.0019

But you are also going to need it just for this course.0022

All right, with each of these, we will start by looking at an algebraic definition,0025

and then exploring what that means in a geometric interpretation, so we can get a sense0028

of how this would look, as opposed to just a bunch of numbers.0031

All right, let's go!0035

First, the dot product: Given two n-dimensional vectors (that is, vectors that have n components,0036

where n will just be some number), u, which will be our first component, u1; our second component,0042

u2; up until the nth component, un...so u1, u2, u3,0048

all the way up until we get to un; and v, which is the same thing: v1, v2,0053

up until vn--so the first component of v, the second component of v...up until the nth component,0058

the last component, since it is just n-dimensional; then the dot product, which is just symbolized exactly as you would guess,0063

a dot between the vectors, is the sum of the products of each pair of components.0070

So, u dot v, which would be the same thing as u1, u2, up until un,0075

dotted with v1, v2, up until vn, would produce u1v10080

(the u1, the first component here, and the first component here multiply together),0086

and then we add that to the second component here and the second component here;0091

and then we add that to all of the ones in between, and then we add that to un,0096

the last one, multiplied by the last one of v, as well.0104

So, we have the first two components multiplied together, the second two components multiplied together...0108

we do this for every one of them, until we get to the last two components, multiplied together.0113

So, we add each of the possible component pairs, each of the component pairs between u and v, for everything that is in the same component location.0119

We multiply each of those together, and then we sum it all up.0126

Notice that the dot product will result in a scalar, not a vector.0129

What we get out of this in the end isn't another vector; it is just a real number.0132

And if we are talking about this, we say u, dot, v: so the symbols are vector u, dot product with, vector v.0137

We just say it as "u dot v," as you have heard me repeatedly say.0144

OK, so let's see an example in two dimensions.0149

If we have (5,2) dotted with (-7,12), then what we do is multiply the first two components, 5 and -7.0150

We have 5 times -7, and then we add that with 2 and 12, multiplied together.0159

5 and -7 gets us -35, plus 24; and finally, we get -11 as our answer when we add it together.0167

All right, it turns out that the dot product of two vectors is deeply related to the angle between the vectors and their magnitude.0175

Now, for us to talk about the angle between the vectors, we can't have them in totally separate places.0182

We have to put them together; so let's say we have two vectors like this and this.0186

Before we can talk about the angle and dot product, we have to put them tail-to-tail.0190

Now, once they are tail-to-tail, we can talk about the angle between them; so right from here to here, where my face is, would be the angle.0193

So, u with θ in between, and then v, is the two angles.0200

Now, it turns out that u dot v is equal to the length of u (remember, if we have vertical bars on either side,0203

that says the length which we get by that magnitude formula, the square root of each component squared and added together),0209

times the length of v, times cosine θ.0217

If we want, we can shift this around to have the formula that makes the angle a little more accessible.0222

Right now, we have cos(θ), but there is other stuff multiplied by it.0226

So, we could solve for cos(θ); we would have cos(θ) = u ·v divided by the length of u, times the length of v.0229

And then, at that point, if we want to find out the angle, we could just take the arc cosine of both sides, the cosine inverse of both sides.0236

And we would figure out what our angle is.0241

For the most part, though, I prefer this formula, because I think it is a little bit easier to remember0243

that u · v is equal to length of u, length of v, times the cosine of the angle between them.0247

All right, it turns out that we can actually prove this formula pretty easily.0252

It is not going to be that hard for us to prove it.0257

But it won't really help us directly understand vectors any better.0259

As such, the proof is still going to be in this lesson, but it has been put at the end, after the examples.0263

So, if you wait until we get through all of the examples, you will get to see the proof, if you are curious about it.0268

If you have a few extra minutes, and you are interested, awesome--I would love to have you check it out.0272

It is really cool to just get the sense of more things in math--all of the proofs.0276

But if you are busy, and you don't have time for it, that is OK, too.0279

It is really...if you had to miss one proof, this is probably the best one to just skip and take in faith.0281

So, don't worry about it if you end up not being able to watch this proof.0286

But if you have the time, and you are interested in proofs, it is really cool, and it is totally something we can actually manage pretty easily.0289

OK, let's interpret this geometrically: we can break up our new formula, u · v = length u times length v times cos(θ),0294

into length u times length v times cos(θ); so we think of it as two different pieces:0302

the length of one vector, times the length of the other vector, multiplied by cosine θ.0308

So, what that gives us: if we look at this, this length v times cosine θ is a way of thinking of the length of v0313

if we projected it onto u--v as a projection onto the vector u.0320

What does that mean? Let's look at these pictures here.0325

We have v here; v is this vector here, so if v is shorter than u, we project it by dropping a perpendicular down to u;0328

and then, what we have is: wherever u went, up until that perpendicular that we just dropped down,0338

that is our length, length v times cosine θ.0344

So, that gives us the length of the projection.0349

We can think about this idea as how far we drop down; once we drop down this perpendicular onto u, we have created a projection of v onto u.0351

It is like you take a flashlight and shine it directly down onto it.0358

It is the shadow that v would cast on u.0361

So, length of v, times cosine θ--well, remember: this is, after all...0365

since it is a perpendicular that we dropped down, we have a right angle in there.0371

So, since we have a right angle in there, and we have θ, it is just basic trigonometry.0374

Cosine of θ times the length of the hypotenuse, length v, will give us the length that it is for that projection--the shadow that we just created.0377

And we can also do this if our vector v ends up being longer than u.0386

Instead of worrying about how long we are on u, it is if u had continued (this dotted line right here),0390

and then we had dropped a perpendicular onto if-it-had-continued, where would we end up being on that continuation?0399

So, we think of u, and then we think of u continuing off forever, and then v drops down onto that, and that gives us our projection of v.0405

So, this gives us another way to think about where this is coming from.0411

It is the length of the projection, multiplied by the length of the vector it is being projected onto.0415

So, we see that u · v is the length of one vector (u in this case), multiplied by the length of the other vector's projection.0421

Our other vector would be v, and its projection was this part right here; that is length v cosine θ.0432

In general, we can interpret the dot product as a measure of how long and how parallel two vectors are.0442

So, for example, if we have two vectors like this, we are going to get a larger dot product the smaller our θ is,0448

because cosine of numbers close to 0 gets us numbers that will be close to 1.0456

Cosine of 0 is 1; that is the absolute maximum.0461

When they are perfectly parallel, that is going to be the largest possible projection that v can make onto u.0464

As it gets less and less parallel, more and more perpendicular, though, cos(θ) is going to become smaller and smaller.0469

This projection that is dropping down is going to become smaller and smaller, until finally, we eventually hit perpendicular,0476

and it drops down, and there is no projection whatsoever.0481

It is like shining a shadow on something pointing straight up; it doesn't cast a shadow at all--there is no projection that comes out of it.0484

So, once they are perfectly perpendicular, we are going to get nothing out of it.0490

But if they are parallel, the more parallel they are, the longer they are, the larger the value that we will get out of the dot product.0493

This brings up a good point--this idea that, when they are perpendicular, when we have cosine of 90 degrees,0501

or cosine of π/2 (degrees versus radians), we end up getting a cosine that comes out as a 0--it goes to 0.0508

So, we will have nothing coming from the dot product, because there will be no projected shadow onto it.0515

From our formula, u · v = length u(length v)(cos(θ)), we see that, if u · v = 0,0521

then it must be the case that cos(θ) = 0, and if cos(θ) = 0, then θ = 90 degrees.0527

So, if we have the dot product of two vectors as 0, we know we have perpendicularity; they have to perpendicular to each other.0535

If u · v = 0, then u is perpendicular to v; and if u is perpendicular to v, then u · v = 0.0545

I also want to point out that there are many equivalent words for "perpendicular."0552

Perpendicular is...sometimes you will hear "normal," or maybe even "orthogonal,"0555

although you normally don't hear that until later math classes, or "at right angles."0560

But all of these might end up meaning the same thing: θ = 90 degrees, or alternatively, θ = π/2, if we are in radians.0564

So, perpendicular, normal, orthogonal, right angles, θ = 90 degrees, θ = π/2--0573

they all mean the same thing, this idea of "perpendicular," something that we are used to from geometry.0579

Don't get confused if you end up hearing one of these synonyms; they just all mean the same thing.0584

You are probably wondering why there are so many synonyms, and it is because it is a really important idea,0588

and it shows up in a bunch of different places; so at different times, different people used different words.0592

And so, we ended up having something like four different ways of talking about this; so, that is why we see so many.0597

Also, I want to point out one other thing: technically, up here, if we have u · v = 0, well, u or v could be the zero vector, as well.0602

If u or v was the zero vector, all 0's in all of the components, then u · v is going to come out as 0, also.0612

We would get u · v = 0 from that, as well.0619

But is our θ really going to be 90 degrees?0621

Well, at that point, if we are just a dot, we are going to say that a dot, that 0 vector, is just going to be perpendicular to everything,0624

because at that point, it is not really sticking out in any direction.0631

So, it seems reasonable to say that it is perpendicular.0634

And it works well with our new idea of "perpendicular" meaning that the dot product equals 0.0637

So, we get around this by saying that the zero vector is going to be perpendicular to all vectors.0642

So, if something is the zero vector, it automatically is going to be perpendicular.0648

But mostly, when we think of this, it is going to work perfectly well.0651

And because of this new thing about saying that the zero vector is always going to be perpendicular, it works out all the time.0656

So, that is just always true; we can think of perpendicular as meaning that the dot product comes out to be 0.0661

All right, let's talk about the cross product now.0667

Unlike the dot product, the cross product will only work in three dimensions.0670

When you are in space, like three-dimensional space, that is when you can use the cross product.0674

It takes two vectors, and it produces a third vector that is perpendicular to both.0678

What comes out of the cross product is something that will be perpendicular to our first vector and our second vector.0684

If u is equal to u1, u2, u3, and v is equal to v1, v2, v3,0690

it is this monster: u cross v equals u2v3 - u3 v2,0694

u3v1 - u1v3, u1v2 - u2v1,0700

where each one of those is multiplied and then subtracted from the other.0704

So, those are our three components.0707

We say u cross v as "u cross v," this vector u cross product with vector v; we just say it as "u cross v" to make it easy.0708

We will calculate a product in Example 1; and basically, this formula is tough to remember, because it is just so many symbols at once.0717

So, there is a pretty good mnemonic for this; but you might not have seen what we are about to use for it.0725

So, we haven't learned about matrices or their determinants yet; but there is a great mnemonic, if you are familiar with it.0731

If you learned about this in a previous math course, or if you go on and watch the lesson on determinants in a little bit--0738

and matrices--there is a great mnemonic for remembering the cross product0743

by using the determinant of a 3x3 matrix and the standard unit vectors.0747

So, if we have u cross v, we can also write that as the determinant of the 3x3 matrix, ijk on the top,0751

then the first vector, u1u2u3, on the second one, and then v1v2v3.0759

At this point, we take ijk; we do a co-factor expansion on ijk, so when we look at i, that will knock out the u1v1 0775.8 and the jk; so we have i along with the determinant, u2u3/v2v3,0766

times the unit vector i, minus...next we do j; j will knock out u2v2; i knocked u1v1,0782

and j and k; j will knock out i and k, and will also knock out u2v2,0793

leaving us with u1u3v1v3.0799

So, we take the determinant of that and multiply that by our j.0802

And then finally, we get to k; k and its cofactor will knock out ij and u3v3.0805

And so, we get u1u2v1v2k.0812

And then, if you take the determinants of each of those 2x2 matrices, multiplying on the diagonal going down,0814

that is the positive; and then subtract on the diagonal going up...so u2v3 - u3v2i,0820

minus u1v3, minus u3v1j,0827

plus u1v2 minus u2v1.0830

That is another way of doing it; it is a pretty good mnemonic; if you are familiar with determinants, it works great.0832

If you are not familiar with this, that all probably didn't make very much sense.0838

So, you can go ahead and watch the later lesson on determinants, or you can just go back to that previous slide and just end up memorizing that formula.0841

There is not really a very easy way to memorize it, other than this mnemonic, which works great.0847

But if you don't know the mnemonic, it is a little bit difficult.0851

Yes, it is really great if you learned determinants, though.0855

And you will learn that later on, so it might be worth just learning it now,0857

so you can have this stuck in your head if you end up having to do a lot of work with cross products.0860

All right, what does this mean geometrically?0864

If you are given two vectors, u and v, the cross product produces a third vector, u cross v, that is perpendicular to both.0868

Say we have some u going off like this, and we have v going off like this.0874

Now, what we have, u cross v, is this third vector that comes out of both that will be perpendicular to both.0881

I think you can see this: so, u cross v would end up being perpendicular to both.0889

It is perpendicular to u, but it is also perpendicular to v.0893

And so, it comes out like that, and we have u cross v.0897

However, how do we tell which way u cross v is going to point?0900

The perpendicular vector can come out like this, but the perpendicular vector could also come out like this.0905

There is nothing wrong with being perpendicular on the underside, as well.0910

So, how do we tell which way you end up going?0913

The trick to this is the right-hand rule: you point the fingers of your right hand in the direction of the first vector.0916

And then, your palm goes in the direction of the second vector; you can also think of it as curling your fingers toward it.0923

And the cross-product's direction is the direction your thumb points.0928

In this one, if this is u and this is v, then you put u...my fingers are along u, and then my palm goes towards v.0931

So, u, v...and u cross v comes out like this, which is exactly what you get from this picture right here.0940

If you do u with your fingers, and then v with your palm, and then bring your thumb out, you will be able to see u cross v coming out in purple there.0947

I really recommend trying this out right in front of you right now, because there is really no way to get this done geometrically,0956

in your head, without actually seeing it visually in front of you.0961

And then, if we wanted to see what v cross u would be, right-hand rule: this is our v; this is our u.0964

So, v goes first; the fingers go in the v direction.0970

And then, u: the palm goes in the u direction, so now our thumb is pointing down.0975

v, u, down; v cross u will go down like that; that is the right-hand rule.0980

Fingers go in the first vector's direction; palm (or fingers curling--either way you want to think about it) goes in the direction of the second vector.0987

And whatever you have with your thumb, that is the direction that your cross-product is going to come out of it.0995

It is definitely worth trying that; try it right now--make sure you end up seeing that you are getting the same thing,1000

that you can do this with your hands, that you can see this, because it is really difficult to visualize1005

purely with your mind; but if you use it with your hands, you will be able to see it very well.1010

This stuff comes up all of the time in physics and engineering--it is really important stuff there.1014

All right, cross-product...now, another way to interpret this geometrically...1019

We talked about the direction, but we haven't talked about how long u cross v is going to be.1023

The magnitude--how long u cross v will be--is equal to the area of the parallelogram that is enclosed by u and v.1027

So, we have u and v; if we continue those out, u is down here, and so we also do a parallel one here.1035

This is parallel to this, and then our v here is parallel to this.1042

So, that makes a parallelogram; the area inside of that parallelogram is how long u cross v will be.1048

So, notice that the more parallel u is to be, the more squished it becomes.1055

If this is our u and v, the more parallel they are, the less area that there is going to be inside of them.1059

They get squished more and more; as we open it up more and more, though, we have this larger area inside of it.1064

So, as we squish it down, as u becomes more parallel to be, the less area it has.1070

The more perpendicular, the more we open it up...we have this wide area.1076

When we are perpendicular, we are going to have the maximum amount of area, because we will be a perfect square.1080

So, the more it opens up, the more perpendicular it is; the more it opens up, the larger the area becomes.1085

With this in mind, we can interpret the length of u cross v as a measure of how long and how perpendicular the two vectors are.1091

If the vectors aren't very perpendicular at all, then we are not going to get much out of the cross product, in terms of its length.1099

If they are really perpendicular, we are going to get a lot more out of it.1106

And of course, we can just make the vectors longer in the first place to increase this area.1110

All right, we are ready for some examples.1114

The first one: A vector, a, equals (2,4,-5); vector b = (-3,1,2); the first thing to do is give the cross product of a and b, a cross b.1116

Then, we want to show, by the dot product, that a is perpendicular to a cross b and that b is also perpendicular to a cross b.1125

So, our first thing to do is just to figure out what a cross b is.1134

We have (2,4,-5), (-3,1,2); so (2,4,-5) is crossed with (-3,1,2).1137

We have this formula here: here is our formula; so for the first coordinate of our outcoming cross-product,1150

it is going to be the second component of the first vector, u2 (u2 would be, in this case, 4),1158

times the third component of the second vector, v3 (so that would be 2 here), so 4 times 2;1165

minus the third component of the first vector (that is a -5), times the second component of the second vector, v2 (that is 1);1171

the same thing is going on; see if you can follow along here...1182

u3 is -5, times v1 is -3, minus u1 (u1, in this case, is 2); v3 is 2,1184

as well; comma, u1v2 (u1 is 2; v2 is 1),1197

minus u2 is 4...v1 is -3; great.1204

We start simplifying this out; we have 8 minus a negative--that cancels out, so we have 8 + 5.1211

-5 times -3 becomes positive 15, minus 2 times 2...minus 4...2 times 1 is 2; minus 4 times -3...they cancel out, and we have addition there,1217

as well; plus 12; simplify that, and we get (13,11,14), so that is a cross b.1229

So, there it equals a cross b; there is our cross-product vector.1243

Now, we want to verify this; we want to show that it is, indeed, going to be perpendicular to both a and b,1249

because we know that the cross-product has to be perpendicular to both of them, so that had better come out.1254

So, if a is perpendicular to a cross b, then that will be true if a dot a cross b comes out to be 0.1258

So, if a dot a cross b comes out to be 0, we know that that is perpendicular by how the dot product works.1267

Remember: that was one of our big realizations about the dot product--that if θ is equal to 90 degrees,1271

if the two vectors are perpendicular to each other, then the dot product of the two vectors always comes out to be 0.1276

a dot a cross b: our a is (2,4,-5); our b...sorry, not our b; we are dotting that with a cross b; a cross b is (13,11,14).1281

2 times 13 is 26, plus 4 times 11 is 44, plus -5 times 14 is -70.1295

26 + 44 becomes 70, so we have 70 - 70; that comes out to be 0, so that checks out.1307

Our dot product came out to be 0, so we know that they must be perpendicular; great.1313

The next one: Show that b is perpendicular with a cross b: so b, dotted with a cross b:1318

b is (-3,1,2); our a cross b that we are dotting with is (13,11,14) (I am running a little bit out of room there);1332

-3 times 13 becomes -39; plus 1 times 11; plus 2 times 14 is 28; -39 + 39...11 + 28...-39 + 39 becomes 0, so that checks out, as well.1344

The dot product of b with a cross b comes out to be 0, so we know that those two vectors must be perpendicular.1361

So, there we are; we have finished that one.1367

All right, the second example: we have u = (5,2,8,k); v = (3,-4,1,3).1369

What is k if u is perpendicular to v? If u is perpendicular to v, then that tells us that u dot v equals 0.1377

Great; so if u dot v equals 0, then we have that (5,2,8,k), dotted with (3,-4,1,3), equals 0.1389

So, we work this out: 5 times 3 is 15; plus 2 times -4 is -8; plus 8 times 1 is 8; plus k times 3 is 3k; equals 0.1406

-8 + 8...they cancel each other out, so we have 15 + 3k = 0; 3k = -15, which gives us k must equal -5.1420

So, there is nothing really difficult there; as soon as we realized that if they are perpendicular,1433

then it must be that their dot product is 0, at that point, we can set something up that we can just solve through simple algebra.1437

The third example: In physics, the work done by a force f over a distance d is defined as force dotted with distance.1443

So, the work, W, is equal to the force vector, dotted with the distance vector.1452

If you push a box with a mass of 20 kilograms with a force of 100 Newtons at an angle of 15 degrees above the horizontal,1457

for 10 meters, how much work have you done on the box?1463

So, the first thing we can do is figure out: All right, what is our force vector in terms of its components?1467

So, we could get the force vector...force equals component stuff; and then we will figure out d = component stuff...1473

well, that will be easy, because it is entirely horizontal; so we will have to use trigonometry to figure out what the force vector is,1481

in its component form, and then we can dot the two together.1486

But that is actually more work than we have to do.1488

All we have to do is remember that we don't need component form at all, because we know that,1490

if work equals the force vector dotted with the distance vector, then, well, u dotted with v is the same thing as length u times length v,1495

times cosine θ, so this is the length of our force vector,1505

times the length of our distance vector, times the cosine of the angle between them.1509

We know what our force vector is--it came out to be 100 Newtons; the force was 100 Newtons.1514

We know what our distance is: we go for a distance of 10 meters.1519

And we know what our angle θ is; do we need the 20 kilograms of mass?1523

20 kilograms of mass actually never shows up for figuring out the work; the mass of the object has no effect on the amount of work that goes in.1526

It is all about the force, the distance that happens, and the angle between those two--how the two interrelate.1533

So, we actually don't need to know the mass of the box at all to figure this one out; it is just a "red herring."1540

So, force is 100 Newtons, times distance (is 10 meters), times cosine of the angle of 15 degrees.1545

We work that out with a calculator: we have 1000 times the cosine of 15 degrees.1556

And that comes out to be 965.93; now, what are the units of work?1560

They told us in the problem that the unit is the Joule, or joules, which is signified with a J.1566

So, we use the unit of J at the end of it.1572

And there we are; there is our work; great.1576

All right, the final example: Prove that u dot u equals the magnitude of u, squared.1579

All right, the first thing to do: we have to have a way of talking about just some general vector u,1585

because they didn't tell us much about u at all.1590

They told us just "vector u"; so we need to be able to talk about what vector u is, in a way that we can actually work with it.1592

Let's just give the components names; we will do it in the same way that has happened in all of the previous stuff,1598

where we have just said that the first component is u1 (so we will make this u1);1604

and then the second component will be u2, and then the third component would be u3,1608

and all the way up until some un, because every vector has to have some specific length.1613

It is not allowed to go on forever; so we will stop at un...n will be just the length of our vector.1619

This is going to be the case for any vector at all; we could put it in this form of first component, second component,1625

up until its last component, which we will say will be its nth location in the thing.1630

All right, so now we have a way of doing this; let's just look at what u dot u is, and then what the magnitude of u2.1634

If they end up being equal, we have proved this thing.1640

So, u dot u would be vector u1, u2, up until un, dotted with u1, u2, up until un.1643

All right, so u1 times u1...well, u1 is just some number, so that is (u1)2,1660

plus...u2 times u2...well, u2 is just some number, so that is (u2)2,1665

plus...this is just going to keep happening, until we get to our final component.1670

un times un...well, un is just some number, as well, so that is (un)2.1673

So, we have u1 squared, plus u2 squared, up until we get to un squared.1677

Now, there is not much we can do to simplify that there; so let's take a look at the magnitude of u2.1682

So, first, what is the magnitude of u? Well, the magnitude of any vector, remember,1688

is the square root of each component squared added together underneath the square root.1693

So, it would be the first component squared, plus the second component squared,1699

up until the final component squared, and all of that underneath the square root.1703

Well, if we square this, then we have the magnitude of u, squared, equals...well, if we square both sides of that equation above,1708

the square root times the square root cancels out, and we just have u1 squared plus u2 squared,1717

up until un squared; well, look: this and this are the same thing.1724

So, since they are the same thing, we have just shown that u dot u is equal to the magnitude of u, squared.1733

Great; the proof is finished.1742

All right, that finishes for the examples; thanks for coming to Educator.com.1744

And we will see you in the next lesson, when we start talking about matrices--all right, goodbye!1748

All right, I think they are gone--everybody who is not actually interested in the proof.1754

So, are you ready for this: Bonus round--here we are, ready for the proof!1757

Dot product formula: let's prove that u · v is equal to the length of u, times the length of v, times cosine θ.1764

The very first thing that you want to do any time you are really trying to think about anything analytically is draw a picture.1770

A picture is always a useful way to think about things.1775

So, we start by drawing a picture: we have u and v, with θ in between.1778

Now, we look at this, and we want to see: is there any way to connect the length of u, the length of v, and θ together?1783

Is there some way to get these things to talk to each other?1788

Do we know any way to say, "Yes, I know these are related"?1791

Well, we look at this for a while, and we say, "Well, I don't see anything yet."1794

But that looks kind of like a triangle; and I know a lot of things about triangles from trigonometry.1798

So, let's say it looks like a triangle without a top; let's give that triangle a top!1803

We draw in the top in this purple color; and we might realize that there are three sides to a triangle;1808

I know an angle...sort of...oh, I can connect the length of u, the length of b, that angle θ, and the length of the top,1816

together with something that we learned in trigonometry: the law of cosines.1823

We might remember this; and if we remember this, we just go back and look up the law of cosines.1827

There is no reason not to just go and look it up.1831

We look up the law of cosines in a book; we refresh ourselves, and we get that the length a2,1833

the side a, squared, is equal to the other two sides squared (b2 + c2),1839

minus 2 times b times c times cosine of capital A, where little a and capital A are the side and then the angle opposite;1846

so little a is the length of one side, and then capital A is the angle opposite that side; so we have:1857

a2 = b2 + c2 - 2bc times cos(A).1863

That is the law of cosines; now, we bring up our vectors picture, and we look at this.1868

Now, one thing before we keep going: how are we going to have this top as u - v?1872

We can see the top as the vector u - v, since it runs from the head of v to the head of u.1877

And think about this: if we take v, and we add that to u - v, well, the -v and the +v would cancel out, and we would be left with just u.1883

So, it must be the case: we can see graphically, through this algebra, that we can get from v to u by using u - v,1891

because it will take away the v and give us the u, and we will manage to get from the head of v to the head of u.1899

Cool; so that is how we have that vector u minus vector v is the way to be able to talk about the top as a vector.1905

All right, using the law of cosines, then we have that the length of u - v, our a, squared, is equal to the length of u, our b,1911

this part right here, squared, plus the length of v, this part right here, c, squared, minus 2 times the length of u,1926

times the length of c, times cosine of the angle between them (cosine of θ).1948

So, we have the length of vector u - v, squared, equals the length of u, squared, plus the length of v, squared,1957

minus 2 times the length of u times the length of v, times cos(θ).1965

All right, it looks like we are getting somewhere: we have some relationships going on.1969

We even have that cos(θ), if we are trying to prove that thing.1972

So, it looks like we are getting there; but we still have this problem, where we don't have any dot products there.1975

So, how can we get u · v to show up in there? We want u · v in there.1980

Well, remember: in Example 4, we just proved, for any vector a, a dotted with itself (a vector dotted with itself)1985

is equal to the magnitude of that vector, squared; so a dot a equals the magnitude of a, squared.1992

Thus, we can swap out each of these magnitude-squareds for a dot a.1998

So, we have that relationship, whatever they are.2004

So, u - v: the length of u - v squared will become the vector u - v, dotted with the vector u - v.2006

u squared: the magnitude of u squared will become the vector u, dotted with the vector u.2014

The magnitude of v squared will become v dot v.2018

So, we have all of these swapping out right here.2022

OK, at this point...we didn't show this technically, but you can prove it to yourself--it is not that difficult:2026

the dot product is distributive, so we can actually distribute using this dot product.2032

u - v dot u - v: well, then, we have u dot u minus u dot v, minus v dot u plus v dot v.2038

u dot u minus u dot v, minus v dot u minus...times a negative again, so plus v dot v.2048

Great; the stuff on the right just stays the same.2060

At this point, we see that we have certain things on the right and the left.2062

So, if we have v · v on both sides, let's just subtract it on both sides.2065

We have u · u on both sides; let's just subtract it on both sides.2068

So, we have - u · v - v · u.2071

Well, notice: u · v is just the same thing as v · u, so we can combine them together.2073

If we have u · v + v · u, then that is the same thing as 2 times u · v.2080

So, if we have - u · v - v · u, then that is the same thing as -2(u · v).2085

So, we have -2(u · v) there on the left; it equals -2 length of u, times length of v, times cosine θ.2090

We have -2 on both sides; divide by -2 on both sides; those cancel out; we have u · v equals the length of u, times the length of v, times cosine θ.2097

Cool; and our proof is finished--that was not too tough.2104

All right, thanks for staying around; I think proofs are really cool.2107

They are really, in my mind, the heart of mathematics--being able to show that this stuff is definitely always true.2110

I think it is awesome; thanks for staying around--I was glad to share it with you.2116

We will see you at Educator.com later--goodbye!2118