Question

Show that angles between any two diagonals $\theta ={\cos }^{-1}\left(\frac{1}{3}\right)$

Answer 1

Directions of $OG,AE,CD,FB$

$OG=1,1,1$

$AE=-1,1,1$

$CD=1,1,-1$

$FB=-1,1,-1$

let angle between $OG$ and $AE$ be

$\therefore \cos \theta =\frac{\left|x\right|+\left|x\right|+\left|x\right|}{\sqrt{1^2+1^2+1^2}\sqrt{{-1}^2+1^2+1^2}}=\frac{1}{3}$

Clearly,

$\cos \theta =\frac{1}{3}$ for any two diagonals.

$\therefore \theta ={\cos }^{-1}\left(\frac{1}{3}\right)$ is angle between any two diagonals.