A vector makes angles $\alpha$, $\beta$ and $\gamma$ with the $x$-, $y$- and $z$-axes. What is $\cos 2\alpha + \cos 2\beta + \cos 2\gamma$?

Explanation

1. 01

   $\cos 2\alpha = 2\cos^{2}\alpha - 1$, so the sum is $2(\cos^{2}\alpha + \cos^{2}\beta + \cos^{2}\gamma) - 3$.

2. 02

   The squares of the direction cosines add to 1, so the sum is $2(1) - 3 = -1$.