Question

The angle between two diagonals of a cube will be

• A. ${\sin }^{-1}\left(\frac{1}{3}\right)$
• B. ${\cos }^{-1}\left(\frac{1}{3}\right)$
• C. variable
• D. None of these

Answer 1

Answer: (B)

Let the cube be of side $a$.
$O(0,0,0),D(a,a,a),B(0,a,0),G(a,0,a)$
then equation of $OD$ and $BG$ are $\frac{x}{a}=\frac{y}{a}=\frac{z}{a}$ and
$\frac{x}{a}=\frac{y-a}{-a}=\frac{z}{a}$, respectively.

Hence, angle between $OD$ and $BG$ is
${\cos }^{-1}\left(\frac{a^2-a^2+a^2}{\sqrt{3a^2}\cdot \sqrt{3a^2}}\right)={\cos }^{-1}\left(\frac{1}{3}\right)$