Exercise 3.5

Write out (v - w)(v - w) using the linearity of the dot product in each of its arguments. What famous law does this establish?

Solution:

The square of the length of (v - w), which is (v - w, v - w) can be expanded to (v, v) + (w, w) - 2(v, w) which is the square of the length of v plus the square of the length of w minus twice the length of v times the length of w times the cosine of the angle between them. This is the law of cosines.