>> Here's how a Dot product is used. It is used -- one use of it is to find the angle between two vectors. The angle between two vectors and here's what I'm talking about, if we have the same two vectors we talked about a moment ago, and I put them on to a coordinate plane and if I position them so that their initial points are both at the origin, then they would look like this. Here's is u and here is v. Now, I know this is u because let's see the components are let's see horizontal and vertical, so it's 5, 2, you see and then v would be 3, 4, okay. Now, it isn't necessarily the case that we have to be talking about the vectors positioned at the origin. They can be positioned just about anywhere on the coordinate plane and we can still talk about the angle between them or the angle relative to those two vectors. And for example if we are thinking about the vectors over here and this would be u then and that would be v, then the angle you see between them, the angle relative to those two would be the angle we find by extending them a bit and this then is the same angle theta that we are looking at up here. So, we are trying to find the size of angle theta. Now, you know from the previous section that one way to do this would be to find the angle associated with the vector v and find the angle associated with vector u and then just take the difference between those two and that wouldn't be very difficult in a case like this, but its process is a little more direct and we have a lot of other uses for the idea of Dot product. Here's how Dot products is used in this process. It turns out that the cosine of that angle between the vectors is the Dot product of the two vectors divided by the product of their magnitudes, the product of their magnitudes. So, let's see the Dot product of u and v. We calculated it before. It's 5 times 3 plus 2 times 4 and remember where that comes from, it's 5 times 3 plus 2 times 4, alright to give this numerator. And then for the denominator, the magnitude of u means the square root of the sum of the squares of the components -- there's a mouthful. But for you it's the sum of the squares of 5 and 2 under a radical. So, it's a square root of 5 squared and 2 squared and for the magnitude of these, same idea, some of the squares of the magnitudes, the magnitudes was 3, 4. So, it's the square root of 3 squared plus 4 squared and then simplifying a bit we find our fraction to be 23 over the square root of 29 times the square root of 25, square root of 25 is 5, so we get this fraction. Now, if the cosine of theta is equal to this fraction, then theta is the inverse cosine of the fraction and we find theta then to be 31 degrees. So, this angle theta then is 31 degrees and the angle, the relative angle between the vectors position like this would also of course be 31 degrees approximately.