The Law of Cosines

The law of cosines is easily derived (one of several ways) by finding the scalar length of the difference vector $\vA - \vB$ .

A-B-theta-vectors.eps

$$\vert\vA - \vB\vert^2 = (\vA - \vB) \cdot (\vA - \vB) = \vA\cdot\vA - \vA \cdot \vB - \vB \cdot \vA + \vB \cdot \vB$$

or (collecting terms and using rules from above):

$$\vert\vA - \vB\vert = \sqrt{ A^2 + B^2 - 2AB\cos\theta }$$

Note that the Pythagorean Theorem is a special case of this rule with $\theta = \pi/2$ .