 While it's possible to treat vectors from a purely algebraic and arithmetic standpoint, it turns out to be very convenient to think about what they represent geometrically. For example, the coordinates of a point in Rn form an n-tuple of real numbers. For example, the coordinates of the point 3, 1, or 5, negative 2. Similarly, we can talk about a vector in Rn, the vector 3, 1, the vector 5, negative 2. If these things look similar, might they be similar? We might proceed as follows. The coordinates of a point tell you how to get to the point from the origin. So to get to the point 3, 1, I need to go 3 units over and 1 unit up. Or to get to the point 5, negative 2, I might go down 2 units and then 5 units to the right. This suggests a geometric interpretation of a vector. While we might give directions like go 3 units to the right and then 1 unit up, we're more likely to point in the right direction and say go that way for some distance. And so this suggests that when we talk about vectors, they are things that have both a direction and a distance. And we might call them directed distances. Well if a vector is a directed distance, then we can represent it using an arrow, showing us the direction of the arrow, showing us the direction of the vector, and the length of the arrow, showing us the distance. And the first thing we might want to do whenever we're given a new type of object is to ask when two of these things are the same. And since a vector has both a direction and a distance, we want to find vectors whose directions and distances are the same. For example, here are two vectors with the same length, but their arrows are pointing in opposite directions, so these two vectors are not the same. Meanwhile, here are two vectors that are pointing in the same direction, but their lengths are different, so again these are not the same. But then there are these two vectors. They point in the same direction, and their lengths are the same, so these two vectors will be equal. It's important to remember that because we're treating a vector as a direction with a distance, the actual location of the vector doesn't matter. We can move them around from place to place as long as we don't change the direction that they're pointing or their length. With these ideas in mind, suppose we want to add vectors. So for example, suppose I want to add u plus v. I can read this as follow whatever direction u tells you to do, and then follow the directions that v tells you to do. Once you follow those directions, you might find a shortcut that will get you to your destination more quickly than following u and then following v. However, there's a problem with what we just did. We didn't just follow the directions u and v, but we included a third set of directions. So that means we haven't found u plus v, but we found u plus something else plus v. So what we need to do is to follow u and then from wherever we end up to follow v. Since the only important thing about a vector is its direction and magnitude, that means we can start anywhere we want. From this point, we'll follow the directions that u gives us, and then we'll follow the directions that v gives us. Now following these directions, u then v is the same as following a single shortcut path. And that path is what we'll call the vector u plus v. How about the vector u plus 2w? So again, we'll start somewhere, then follow u, and then I want to follow the directions that w gives me twice so I can go to w. And again, rather than following u plus w plus w, we might see the shortcut path. And that shortcut path is what we'll call u plus 2w. So let's find a few vectors. We'll find the vectors op, that's the vector that gives us the directions from the origin to the point p, where as a reminder, we have an arrow pointing from o towards p. Let's also find the vector pq, that's the vector that gives us the directions from the point p to the point q. And finally, let's find the vector qp, those are the directions for getting from the point q back to the point p. To find these vectors, remember that we specify the location of a point in space by the use of multiple axes, so the vector op is going to tell you how to get from the origin to the point p. So to get to the point p, we have to go three units along the x-axis, one unit parallel to the y-axis, and then down two units parallel to the z-axis. And so our vector will be 3, 1, negative 2. Notice that the components of the vector from the origin to the point are the same as the coordinates of the actual point. And that's going to be true in general because the coordinates do tell you how to get to the point from the origin. What about the vector pq? If we compare the coordinates, we see that p has x-coordinate 3, while q has x-coordinate 5, which means that if we want to get to q, we have to go two units farther out, parallel to the x-axis. Next, p has y-coordinate 1, while q has y-coordinate negative 2, which means we're going to have to decrease our y-coordinate by 3. We're going to have to go negative 3 units parallel to the y-axis. And finally, looking at our z-coordinate, p has z-coordinate negative 2, q has z-coordinate 1, which means we'll have to go positive 3 units parallel to the z-axis. And so our directions for getting from p to q are going to be to go 2 units parallel to the x-axis, negative 3 units parallel to the y-axis, and 3 units parallel to the z-axis, which corresponds to the vector 2, negative 3, 3. Finally, if we want to find the vector qp, we should just reverse those directions and go negative 2 units parallel to the x-axis, positive 3 units parallel to the y-axis, and negative 3 units parallel to the z-axis. And so our vector qp will be negative 2, 3, negative 3. Our work on this problem suggests the following definition. If I have two points A and B in R, N, then the components of the vector A, B, will be the difference between the coordinates of B and the coordinates of A. So that'll be B1 minus A1, B2 minus A2, and so on.