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)

${\cos }^{-1}\left(\frac{1}{3}\right)$

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
${\cos }^{-1}\left(\frac{a^2-a^2+a^2}{\sqrt{3a^2}\cdot \sqrt{3a^2}}\right)={\cos }^{-1}\left(\frac{1}{3}\right)$
Note: Students should remember this question as a fact.