 A vector has both magnitude and direction, and finding the magnitude is pretty easy. The direction of a vector is a little bit more complicated. In fact, it's so complicated we won't try to define the direction of a single vector, but we can talk about the angle between two vectors. So let's try to find the angle between the vectors 1, 4, and 2, 5. Now, suppose we treat both vectors as giving us directions to a point from, well, let's make it from the origin. And remember, if it's not written down, it didn't happen. Let's go ahead and label the origin. So 1, 4 will take us to the point with coordinates 1, 4, and 2, 5 will take us to the point with coordinates 2, 5. To find the angle between the two vectors, the thing you might notice is that they form two sides of a triangle. And that third side is going to go from 1, 4 to 2, 5, and this corresponds to the vector 1, 1. Now, if I want to find the angle between two sides of a triangle, I can use the law of cosines, which requires me to find the length of all three sides. Well, that's just the magnitude of the three vectors 1, 4, 2, 5, and 1, 1. So finding these magnitudes, and if it's not written down, it didn't happen. Let's go ahead and put those in our diagram. These correspond to the lengths of the three sides of our triangle. And we're ready to use the law of cosines. So using the law of cosines, filling in our values, and solving for the angle we find, and so we find the angle between the vectors is about 8 degrees. What about vectors in three dimensions? Well, again, if we treat these as directions from the origin, they form two sides of a triangle. One side goes from the origin to 1, 4, negative 1, and we find the length of that side is the magnitude of the vector. Another side goes from the origin to 2, 5, negative 3, and the length of that side is the magnitude of the vector. Now, we do need the vector joining those two endpoints. That third side is going to go from 1, 4, negative 1 to 2, 5, negative 3. So remember, a vector tells us how to get from one point to another, and so the vector joining the two endpoints will be, and its magnitude will be, and again, we have three sides of a triangle, and we want to find the angle between two of the sides, so we use the law of cosines, and we find the angle will be... Now, this seems to be a lot of work, so you might wonder if there's a better way to compute the angle between the two vectors, and the answer is, there isn't. However, the computations we're doing are always the same. We find the magnitude of the two vectors, find the magnitude of the vector joining the two, use the law of cosines, solve for cosine, and then find the inverse cosine. The important thing here is that we're always doing the same thing, and any time you do exactly the same sequence of steps, you can produce a formula to do all the steps at once. And so this means we can do all of these steps at the same time and produce a formula, and we'll take a look at that next.