What is the acute angle between two body diagonals of a cube?

Explanation

1. 01

   With side $a$ and edges along the axes, two body diagonals run from $(0, 0, 0)$ to $(a, a, a)$ and from $(a, 0, 0)$ to $(0, a, a)$: direction ratios $1, 1, 1$ and $-1, 1, 1$.

2. 02

   $\cos \theta = \frac{\lvert -1 + 1 + 1 \rvert}{\sqrt{3}\sqrt{3}} = \frac{1}{3}$, so $\theta = \cos^{-1}\frac{1}{3}$.