WEBVTT mathematics/multivariable-calculus/hovasapian
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Welcome to educator.com.
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Welcome to the first lesson of multivariable calculus at educator.com.
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Multi variable calculus is an extraordinary branch of mathematics.
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Those of you who are coming to this course, you have already come from a course in regular calculus.
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What we are going to do is, we are going to take the power of calculus, and we are going to move from one dimension.
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We are just going to move up, to two dimensions, to three dimensions, and as it turns out, any number of dimensions.
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That is what makes all the things that we are going to learn really, really exciting.
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We are not limited to two and three dimensions, in general.
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We are going to be doing most of our work in two and three dimensions, because we want to visualize things.
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We are used to playing in space and things like that, but the results are valid for any number of dimensions.
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So, let us jump right in and see what we can do. Welcome again.
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We are going to start off with just some normal basics.
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We are going to talk about points and vectors just to get ourselves going.
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A lot of the stuff that you may have seen before, if you have not seen it before, it is reasonably straightforward.
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Let us start off by just defining a point in space.
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A point in space, can be identified... Actually, I am sorry, let us start with a point on a plane, not in space.
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We will work ourselves into 3 space in just a minute.
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A point on a plane can be identified with two numbers.
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One number for the x-coordinate, and one number for the y-coordinate.
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This is nothing that you have not been doing for years and years and years.
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Just the normal Cartesian coordinate plane.
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You have an X axis, and a Y axis, and when you choose a point in that plane you need 2 numbers to define where it is in that plane.
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Let me just go ahead and finish writing off the y-coordinate.
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So, a quick picture, you are going to have something like this.
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This is the Cartesian coordinate plane and let us take a number like that.
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So, this point might be (6,2).
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In other words you are 6 in the x direction, this is the X axis only, the X axis is horizontal, Y axis is vertical, and 2 units in the Y direction.
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That gives us a point in space.
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Now, because we need two numbers to identify that point, we call that a two-dimensional space.
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So, whenever you hear the word dimension, do not freak out, it is just a fancy word for number.
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When we talk about a 17 dimensional space, it just means that in order to identify a space, or a point in that space, we need 17 numbers.
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That is all it is, dimensions, numbers, they are synonymous.
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Okay, so now let us actually talk about a point in space, the space that we think of as the space that we live in, the space that we know, 3 space.
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So a point in 3 space, actually let us not identify it as 3 space yet, let us go nice and slow.
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A point in space can, can be identified with 3 numbers.
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Again, one for X, one for Y, and one for z, we generally do x-coordinate, y-coordinate, z-coordinate.
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Let us go ahead and draw what this looks like.
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This is going to be a standard coordinate system called the right-hand coordinate system for visualizing points in three-dimensional space.
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What we have is something like this, slightly different.
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This right here, this is actually the z axis, and this one that is slanted, this is actually going back and coming forward.
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So, this z-axis and this Y-axis are actually the plane of the paper.
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This right here is actually the X axis so, what we have done is we have taken the XY, we have flipped it down, and we have added a z-axis to it.
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If you have some point like, I do not know, let us put a point over here, and let us call this point (3,3,2).
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We have three numbers identifying that point.
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That means that we have moved three units in the X direction, we have moved three units along the Y direction, and we have moved two units along the Z axis.
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That is it, three-dimensional space, because we need three numbers to represent that point.
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Now, analogously we can talk about any number of dimensions.
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We can talk about five space, six space, seven space.
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It just means we need 5, 6, 7, you know, numbers to identify a point in that space.
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The behavior is exactly the same, even if we cannot visualize it.
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In other words we are kind of limited to two dimensions and three dimensional representations theoretically,
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But algebraically, we can just do, for example, if we have a space which is (4,7,6,5,9), this is a five dimensional space, this is a point in a five dimensional space.
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We cannot visualize it, and we cannot draw it, but we can work with it algebraically.
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Let us go ahead and talk about a notation briefly, a notation for a space of a given dimension.
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This is a notation that we will use reasonably regularly.
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We will start to use it more often when we talk about functions from say a two-dimensional space, to a one-dimensional space.
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Taking a number that has, you know, two coordinates, and mapping it and doing something to those numbers, and coming up with an answer that ends up in a different space.
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This notation that I am going to introduce, you will see it more later on, but I do want to introduce it right now.
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So, R is the symbol for the real number system.
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You will have R represents 1 space, and so when we say the space R, we are talking about one dimensional space.
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In other words, a line.
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Basically, you need just one number.
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The reason we use R is because, you notice 2 space is 2 numbers, 3 spaces is 3 numbers, 5 spaces has 5 numbers,
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Well you are taking these numbers, the 6, the 2, the 3, the 3,
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You are taking them from the set of real numbers so R2, when we put a little two on top, that is two space.
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It is saying that we are taking, well, this two tells us that we need two numbers, one from the real number system, another one from the real number system to identify that.
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So R2 is 2 space, it is the Cartesian coordinate plane.
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Oops, we are starting to get these crazy lines again.
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Let us see if we can write a little bit slower to avoid some of that.
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The real number system, actually, let me go ahead and move to the next page here.
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So, R is 1 space, R2 is 2 space, and R3 is 3 space.
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When we talk about the space that we live in, we are actually talking about R3.
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In other words, three numbers from the real number system to identify a point in space,
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We will be using that more when we talk about functions again.
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OK, these points in space are actually individual objects.
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You can think of them just like numbers.
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For example, if I said I have the number 14 and the number 17,
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These are individual mathematical objects, these points in space are also just single individual mathematical objects.
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The difference is we need more than one number to actually describe them.
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These numbers are the coordinates.
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As you will see in a minute, these points in space which we are actually going to start to call vectors in a minute, they are individual objects, and we treat them like individual objects that you can treat just like numbers.
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It is just that they happen to be made up of more than one thing.
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It will make more sense as we start to do the problems.
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OK, so now let us go ahead and talk about vectors.
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We often call points in space, vectors, and in fact, that is probably going to be the primary term that I use.
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It is just so ubiquitous, so common in mathematics to speak about a point in space as a vector.
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We will go ahead and get into the geometry a little bit, but just know that it is just another term for a point in space, that is all it is.
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So (1,-7) a point in two space is called a 2 vector, and exactly what you might think,
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The point (3,-4,6) is a 3 vector, that is it.
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The reason we call it a vector is, again, some of you people might have seen vectors in physics or another math course, perhaps some of your calculus courses actually talked about vectors, I do not know.
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Occasionally they do.
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We call them vectors because you can actually think of a point in two ways.
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You can think that it is just a point in space, or you can think of it as an arrow from the origin to that point in space.
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Let us go ahead and draw it out.
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The geometry would be something like this.
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So, if I had some random point, so this is the x-y plane, and if I had the point (6,2), that is a point.
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You know 6 along the X axis, 2 along the y-axis,
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But it also represents an arrow, an actual direction from a point of origin in this case, the origin, in that direction.
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This is called a vector and it is a directed line segment.
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It is not just a point in space, it actually gives us a direction and that is why we call it a vector.
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So there are two ways of looking at it.
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In general, we will be using these arrows in 2 and 3 dimensions, mostly we will be working in two-dimensions when we do the examples,
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Simply because we want to be able to geometrically help us see what is going on algebraically.
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But geometry is not mathematics, algebra is mathematics.
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We will be working with coordinates, and we will be using these arrows to help us sort of see what is going on.
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Even then, eventually we will have to put the arrows aside, we want to work with just the algebra.
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Geometry helps, it will help guide us.
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It will give us a little of a physical intuition for what is happening.
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The algebra is the mathematics,
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Those are the skills that we want to develop and that we want to nurture, algebraic manipulation, not just geometric intuition.
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Let us see, the three vector, so algebraically, we had the point (6,2), geometrically, we have this arrow.
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Let us see, how are we going to symbolize our vectors?
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We are actually going to have 2 or 3 different ways of symbolizing them.
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I will try to be as consistent as possible, but I am going to at least introduce three different symbols that I am going to use for vectors.
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The reason that we do that is, as mathematics becomes more complex, becomes more involved, the symbolism needs to give more information.
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Often times, you might need more than one symbol to talk about a specific concept, because sometimes it is easier to think about it this way, sometimes it is easier to think about it this way.
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But again, we will talk about them, we will not just drop the symbols, we will talk about what we mean so it should be reasonably self consistent.
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So, vectors will be symbolized as follows.
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We will often write a vector as a capital letter a.
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For example, (3,1,2), sometimes we will write a capital letter with an arrow over it, so (3,1,2) is the same thing.
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Sometimes we will use a small letter with an arrow over it.
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In this lesson ,and probably the next couple of lessons, I will probably using this notation more than any other.
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So, again (3,1,2) they are all the same thing, this is just a vector and actually a point in space.
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Which point in space? The point (3,1,2) three along X, one along y, two along the z.
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Now, let us talk about adding vetors.
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We will add vectors the same way that we add numbers.
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Or I should say vectors, like I said are individual objects, and you can actually add them like you do numbers, there is an addition and it is actually perfectly analogous.
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Let us write out the definition.
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So adding vectors, let a = vector (a1,a2,a3), so these are just different components, I am using variables for them, and we will let the vector B be (b1,b2,b3).
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Well, the vector A + B is equal to, exactly what you might think, a1 + b1, I just add the corresponding components, and I get a point, in this case, three space,
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So, a1 + b1, a2 + b2, a3 + b3, so that is it, I take a point in space, I take another point in space, I can actually add those points in space by adding their individual coordinates, and I end up with another point in space.
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That is what is important I start with two objects, a point in three space,
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I do something to them and I end up with an object in three space.
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That is actually very, very important.
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We stayed in the same set, which is not always the case.
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Later we might be jumping from 3 space to 5 space, from 1 space to 2 space.
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We can really do that, and that is what is really beautiful about multivariable calculus.
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We are no longer constrained just to work a function from x to y.
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You take a number, a function like x squared, you put some number in it, you get another number out, number to number.
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Now we are working from spaces in one dimension, to spaces in another dimension. Very, very exciting.
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Let us do an example, OK, example 1.
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We will let the vector a be (4,-7,0).
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Again, most of our examples are going to be in two and three dimensions to make the mathematics approachable, but it is true in any number of dimensions.
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In fact, one of the culminations of this multivariable calculus course is when we prove, later on.
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We are going to be discussing Green's Theorem and Stoke's theorem.
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Green's Theorem and Stoke's Theorem are just generalizations of the general theorem of calculus to 2 dimensions and 3 dimensions, respectively.
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As it turns out, the fundamental theorem of calculus is true in any number of dimensions.
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One of the most beautiful theorems in mathematics is something called the generalized Stoke's theorem.
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That is exactly it, as it turns out, everything that you learned is not only true of the space that you know, one dimension, but is true in any number of dimensions -- that is extraordinary.
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So (8,3,-6) is our other point, so let us go ahead and add.
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The vector C, which is the sum of the vectors a + b is equal to, well, 4 + 8, we get 12, -7 + 3, we get -4, and 0 + -6, which is -6.
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That is it, that is our answer.
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OK, so now let us list some properties of vector addition.
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Again these properties you know of but we just want to be formal about it.
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So vector a plus vector b, if I do that addition, and then I add another to them, it turns out that it is associative.
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I can add them in any order, it is vector a plus the quantity, vector B + vector C.
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This is just like normal numbers, we are just working component wise.
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We are talking about an object, a vector, an actual mathematical object.
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Okay, the vector sum A + B is equal to the vector B + A,
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It does not matter what order I add them, the vector sum is commutative.
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We are going to identify something called the zero vector.
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The zero vector is the vector that has 0 as all of its components.
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For example, the zero vector at three space would be (0,0,0).
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We are definitely going to distinguish between the zero vector and all of these individual zeros.
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The zero vector, I will draw a line over it, they are actually two different objects.
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One is a point in space, that point in space has coordinates all 0.
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That is the zero vector, remember a vector is an actual object.
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Now, if a is equal to, let us say (a1, a2, a3), we will just work in three space,
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Then -a = (-a1, -a2, -a3).
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When we negate a vector, all we do is we just negate each component of that vector.
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Last of all, if a = (a1, a2, a3), then some constant C × the vector a = well, c × a1, c × a2, c × a3.
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All we have done is take some constant like seven times a given vector.
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That means we multiply each component by 7, that is all we are doing, essentially we are just distributing.
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There is nothing strange going on here.
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Let us do another example very quickly. Example number two,
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We will let a equal (7,radical(6),2) and you see, real numbers they do not have to be integers they can be any number at all.
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They can be π, they can be e, it could be anything.
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I will let the vector b be equal to (2,2,0).
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Let us do a - b the vector a - the vector b.
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That is equal to the vector a, plus the vector -B, because we know that there is no such thing as subtraction in mathematics.
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There is actually only one operation in mathematics that is addition.
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Subtraction is addition of an inverse number, multiplication is actually just multiple additions.
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Division is just the inverse of multiplication which is ultimately based on addition.
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So there is really only one arithmetic operation or mathematic operation in mathematics it is addition.
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Everything else is derived from it, so when we talk about one number - another number, we are talking about something plus the inverse.
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In this case, the vector + the negative of the other vector.
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A + -B that is equal to 7 + -2 which is 7-2, that is equal to 5, radical(6) + -2, we just write sqrt(6)-2,
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That is a perfectly valid number, we will leave it like that, and 2-0 = 2,
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Well that is it, a minus b is that.
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How about if we do 7 × a, like we said, well, 7 × a, a is (7,sqrt(6),2).
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That just means that we take 7 and multiply each of those components by 7, so we get 49, 7 radical(6), and we get 14, that is it.
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Note, sometimes, notice we have been writing our vectors as points in space horizontally.
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Listing the numbers horizontally component wise.
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Sometimes we are actually going to list them vertically, it is just a notational device.
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Later on we will begin to work with matrices.
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So, in multivariable calculus, you will see why sometimes we write them vertically.
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Just know it is the same thing when we write them this way or this way.
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So, sometimes we will write the vector a vertically.
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If I had the vector a, which is, let us say, a four vector (1,7,4,2), a point in four space that is going to be equivalent to (1,7,4,2).
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The only thing you have to watch out for is you want to retain the order -- (1,7,4,2) is a specific order.
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You cannot just write (1,2,7,4), it is not just a set of numbers.
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It is a list of numbers in a specific order, so vertical, horizontal, it is just a question of what is convenient for us at the time.
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Okay, so 2 more properties.
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Two more properties concerning vector addition.
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The first one says that if I take a constant, if I take two vectors a + b, and if I add them first, and I multiply them by a constant, their sum is the same as, if I can distribute this constant over each and then add them, a constant times the sum of two vectors is equal to the sum of a constant times each individual vector.
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I can just distribute the constant.
00:25:02.000 --> 00:25:05.000
I can also do it if I have two constants instead.
00:25:05.000 --> 00:25:10.000
Multiplied by a given vector I can distribute the vector over the constants.
00:25:10.000 --> 00:25:17.000
So it would become C1 × a plus c2 × a.
00:25:17.000 --> 00:25:28.000
That is it, you already know all of these common properties of the real numbers, all of the behaviors are exactly the same.
00:25:28.000 --> 00:25:34.000
Let us see what we have, we have done the properties.
00:25:34.000 --> 00:25:42.000
I want to talk real quickly what we mean geometrically when we multiply a vector by a constant.
00:25:42.000 --> 00:26:19.000
Geometrically, multiplication of a vector by a constant, is just changing the length of a vector, and/or the direction of the vector.
00:26:19.000 --> 00:26:30.000
Let us take a standard vector, say we have this vector right here, a.
00:26:30.000 --> 00:26:34.000
Let us say a is the vector, the point in space, (3,2).
00:26:34.000 --> 00:26:40.000
Again, we are going to be working in 2 space where we can actually visualize these things easily.
00:26:40.000 --> 00:26:42.000
Well if I said 3a, you know what the algebraic value of 3a is.
00:26:42.000 --> 00:26:50.000
It is just 3 × 3 which is 9, and 3 × 2 which is 6, which is (9,6).
00:26:50.000 --> 00:26:54.000
So the new point is (9,6).
00:26:54.000 --> 00:26:57.000
All you have done to the vector is multiplied its length by three.
00:26:57.000 --> 00:27:03.000
That is 2 that is 3, so this is the point (9,6).
00:27:03.000 --> 00:27:14.000
Let us see if we did -2a, now we have not only increased its length by 2, but we have actually changed its direction.
00:27:14.000 --> 00:27:21.000
We have reversed it, so in this case - 2a algebraically, well -2 ×3 is -6, -2×2 is -4.
00:27:21.000 --> 00:27:33.000
Now, we have this is -1a, and this is -2a, so now our point (-6,-4) is right there.
00:27:33.000 --> 00:27:40.000
That is all you are doing.
00:27:40.000 --> 00:27:43.000
Given a vector, negating that vector means reversing the direction of that vector.
00:27:43.000 --> 00:27:48.000
So, if it points this way, it is going to point in the opposite way 180 degrees.
00:27:48.000 --> 00:27:53.000
If you multiply a vector by a constant, if that constant is bigger than one, you are lengthening the vector, but you are keeping the direction.
00:27:53.000 --> 00:27:56.000
Or, it could be smaller than one and you are taking the vector and you are shortening it.
00:27:56.000 --> 00:28:00.000
Say multiply this by 1/2, now the vector becomes (3/2,1).
00:28:00.000 --> 00:28:10.000
So again, you are just changing the length, that is all you are doing.
00:28:10.000 --> 00:28:19.000
Okay, so this was just a basic introduction to points in space and vectors, and how we are more often than not going to just be speaking of points in space as vectors, and with a little bit of instruction in notation.
00:28:19.000 --> 00:28:22.000
So thank you for joining us here with our first lesson from multivariable calculus,
00:28:22.000 --> 00:28:23.000
We will see you next time, bye-bye.