The Dot Product and Cosine

Next, we'll show that the dot product of two vectors is the product of their lengths and the cosine of the angle between them.



First consider two arbitrary vectors A and B. We'll form a triangle with these two sides and a third side connecting the ends of A and B, which we'll call C. Also, let θ be the angle between sides A and B



Notice that C = A - B, and so



These simplifications rely on properties of the dot product that I worked out last week. Also, since the dot product of a vector with itself is equal to its squared length, we have



and according to the Law of Cosines



Combining these two we can see that