 One extremely useful concept in mathematics is the idea of a vector. Now, we can start by looking at the Cartesian product given two sets A and B. I can form the Cartesian product A cross B, and this is going to consist of a set of ordered pairs where the first component is drawn from the first set and the second component is drawn from the second set. And the elements of the Cartesian product can also be referred to as vectors. Now, a little bit of notation here, if A and B happen to be the same set, then I can express the Cartesian product A times A in pretty much the way we would expect it to be written, A raised to power 2, and I can extend this to as many sets as I want. Now, most commonly, we introduce vectors in terms of our set being the set A of real numbers. And so, if A is a set of real numbers, then a vector in R to the K that can be interpreted as a direction in K dimensional space. So, given some point, we have the following ideas. We have the point itself, wherever that point is located. We have the vector OP giving the direction from the origin of our Cartesian coordinate system to wherever the point P is located. Now, that's just the geometry of the situation. So, I can also talk about the coordinates of the point itself, which we can express in our standard form as an ordered K tuple. And finally, I can talk about the vector itself, which I can express as an ordered K tuple. Now, if you're paying attention, you might notice there is a little bit of a problem here in that we're using exactly the same notation to indicate the point as we are to indicate the vector. And here's a rough guideline. Useful ideas tend to get reinvented by many people in many different ways. And one of the ways you can tell that that idea in mathematics is useful is how many different ways we have of expressing the same basic concept. So, we might express a vector this way, which makes it a little bit confusing because it looks like we're describing an ordered pair at the location of a point. So, what else can we do? Well, we might also describe it using square brackets to enclose the coordinates. Or we might use these angle brackets to enclose the coordinates. And, well, why not? Let's see if we can find another way of expressing this. One other thing we might do is we might then express it as a column vector in a matrix. And all of these express exactly the same way of expressing the same vector. Now, for typesetting purposes, we're going to try and write our vectors in this form as often as possible, mainly because there's almost no reason why we'd ever be concerned about the coordinates of a point. So, let's take a look at this. Given two vectors, I can define the vector addition as follows. So, I have my two vectors, my components are v1 through vk, and my second vector, w1 through wk. And my vector sum, vector v plus w, is going to be the component-wise sum of the individual vectors. And in a fit of mathematical originality, we call this component-wise addition. Now, given any real number, we can define scalar multiplication by that real number as a times the vector v. And all we're going to do here is we're going to take our real number and we're going to multiply every component of the vector by that real number. And so, by extension, we're going to get a couple of other things that are useful to have. First of all, what I can write as negative vector w. I can view that as negative one scalar multiplied by the vector w. I can define the subtraction of two vectors in almost the same way I define addition of two real numbers, which is to say I'm going to add the negative of the other vector. And given two real numbers and two vectors, I can talk about the linear combination of those two vectors. And again, I can extend this idea of linear combination to as many vectors as I want. So let's take a look at an example. So here I have two vectors v and w, and I want to find a number of different quantities. So let's see, three times the vector v. What I'm going to do is I'm going to take each of the components of the vector v and multiply them by three. So my new vector three times three, three times one, three times four. And I can multiply those out. That's nine, three, twelve. Likewise, two w, I'm going to take each of the components of the vector w and multiply them by two. And after all the dust settles, I have the vector two, ten, negative sixteen. And then finally, our linear combination, three v plus two w. I'm going to add two vectors together. So I'm going to add them component-wise, nine plus two, three plus ten, twelve plus negative sixteen, and that gets me my linear combination.