 Remember, the coordinates of a point are a set of directions that tell you how to get to the point from the origin. So in two dimensions, the first x-coordinate specifies the horizontal distance, and the second y-coordinate specifies the vertical distance. In six dimensions, it doesn't really matter. Each coordinate specifies distance in a particular direction. So we might try to locate in two dimensions, the point 5, 3, and in three dimensions, the point 2, negative 4, 6. So the coordinates 5, 3, in two dimensions tell us we can get to the point by travelling five units horizontally, right, and then travelling three units vertically, upward. Now if I'm in three dimensions, the coordinates 2, negative 4, 6 in three dimensions tell us we can get to the point by travelling two units towards. That's if we use our standard orientation in the first octant, and then we'll travel four units to the left, since our y-coordinate is negative, and then finally our z-coordinate is 6, so we'll travel six units upward. But what if we want to get from one point to another point? Well, we could navigate from the first point back to the origin using the coordinates of the first point, and then navigate to the second point from the origin again using the coordinates of the second point. So let's say I want to get from the point 3, 5 to the point 1, 4. Now, since the point 3, 5 is the point you get to by travelling from the origin and going three units to the right and five units upward, we can reverse these steps to get back to the origin. And so we'll go five units downward and three units leftward. Now, once we're at the origin, we can get to the point 1, 4, because its coordinates tell us how to get there. And so we'll go one unit to the right and four units upward. And so altogether our sequence of steps will be go five units downward, go three units leftward, then one unit to the right and four units upward. Now, in some places that's how you navigate. You get back to a starting point and then navigate onward from there. It's sort of like driving in Boston. But having to return to the origin does seem a bit inefficient. And so we might ask the question, maybe we can do better. And in fact, it's important to remember always ask that question. Always ask, is there a better way? So let's try that again. Let's describe how to get from the point 3, 5 to the point 1, 4. And so we might notice that if we go to the left by 2, our x-coordinate will decrease by 2 and we'll be at 1, 5. Now, while you might say, OK, well, sure, but why would we want to do that? And the reason we'd want to do that is that makes our x-coordinate the same as where we want to be. Well, how do we make our y-coordinate the same as where we want to be? Well, currently, our y-coordinate is 5. We want it to be 4. And so if we go down 1, our y-coordinate will decrease by 1 and we'll be at 1, 4. And so if we put our directions together, we can follow them. We're going to go left 2 units to 1, 5 and then down 1 unit to 1, 4. So for example, let's describe how we can get from 2, 5 to 1, 7. So our x-coordinate needs to decrease, so we'll go left 1 unit and that takes us to the coordinates 1, 5. Our y-coordinate needs to increase, so we'll go up 2 units to 1, 7. And this leads us to an important concept. A vector is a set of directions that describe how to get from one point to another point. For example, go left 1 unit, go up 2 units. And it's helpful to compare these with coordinates. Coordinates are a set of directions that describe how to get from the origin to a point. Go left 1 unit, go up 2 units. As you can see, there's not a lot of difference between the idea of a vector and the idea of a coordinate. Really, a vector is a generalization of the idea of a coordinate. In a coordinate, our starting point is always going to be the origin. In a vector, we can start anywhere. And so the only real difference between vectors and coordinates is that a vector can start anywhere. Now to distinguish between vectors and coordinates, we typically use angle brackets. So the angle bracket negative 1, 2 is a vector. Meanwhile, parenthesis negative 1, 2 is a set of coordinates. But in both cases, they specify the same directions. Go left 1 and up 2. And the only difference is that with this set of coordinates, we know where we're starting. But with the vector, we know that these are the directions we're getting from some point to another point.