Let's take the cosine rule from algebra, which you'll remember probably, vaguely from school. I might have said, if we had a triangle with sides a, b, and c, then what the cosine rule said was that c squared was equal to a squared plus b squared minus 2ab times the cos of the angle between a and b, cos that angle theta there. Now, we can translate that into a vector notation. We call this vector r here, and we call this vector s here. Then this vector will be minus s plus r, so that vector will be r minus s, minus s plus r. So we can say that c squared was the modulus of r minus s squared, and that would be equal to the modulus, the size of r squared plus the size of s squared minus 2 mod r mod s cos theta. Now, here's the cool bit. We can multiply this out using our dot-product, because we know that the size of r minus s squared is equal to r minus s dotted with itself. Now, that's just that, and we can multiply that out and then we'll compare it to this right hand side here. So r minus s dotted with r minus s. Well, that's going to be, if we need to figure out how to multiply that out, that's going to be equal to r dotted with r, and then take the next one, minus s dotted with r, minus s dotted with r again, if you take minus s and that r. Minus s dotted with r again, and then minus s dotted with minus s. So that is, we've got the modulus of r squared here, and we dot r with itself, minus twice s dotted with r, and then minus s dotted with minus s. Well, that's going to be the size of minus s squared which is just the size of s square. Then we can compare that to the right-hand side. When we do that comparison, compare that to the right-hand side, the minus r squareds are going to cancel, the r squareds even, the s squareds are going to cancel. So we get a result which is that minus twice s dotted with r, is equal to minus twice modulus of r, modulus of s, cos theta. That is, and then we can lose the minus sign, minus signs will cancel out just multiply through by minus 1. Then the 2s we can cancel out again. So we can say that the dot-product r.s, just to put it in a more familiar form is equal to mod r mod s cos theta. So what we found here is that the dot product really does something quite profound, it takes the size of the two vectors. If these were both unit length vectors those will be 1 and multiplies by cos of the angle between them. It tells us something about the extent to which the two vectors go in the same direction, because if theta was 0 then cos theta would be 1, and r.s would just be the size of the two vectors multiplied together. If the two vectors on the other hand we're at 90 degrees to each other, if they were, r was like this and s was like this and the angle between them, theta, was equal to 90 degrees, cos theta, cos 90 is 0, and then r.s is going to be, we can immediately see, r.s is going to be some size of r, some size of s, times 0. So if the two vectors are pointing at 90 degrees to each other, if they what's called orthogonal to each other, then the dot product it's going to give me 0. If they're both pointed in the same direction, so s was like that and the angle between them is nought. Cos of nought is equal to 1, and then r.s is equal to the mod r times mod s, just the multiplication of the two sizes together. Fun one, last fun one here, is that r and s are in opposite directions. So let's say s was now going this way, and the angle between them was a 180 degrees cos of 180, 180 degrees is equal to minus 1. So then, r.s will be equal to minus the size of r times the size of s. So what the dot product here really does with this cos, it tells us when we get the minus sign out that they're going in opposite directions. So there's some property here in the dot product, we've derived by looking at the cosine rule, that we've derived here, when the dot product's 0 they are 90 degrees to each other, they're orthogonal, when they go in the same way we get a positive answer, when they're going more or less in opposite directions we get a negative answer for the dot product.