Question

The $\text{cosine}$ of the angle between any two diagonals of a cube is

• A. $\frac{1}{3}$
• B. $\frac{1}{2}$
• C. $\frac{2}{3}$
• D. $\frac{1}{\sqrt{3}}$

Answer 1

The correct option is A $\frac{1}{3}$

Let a cube with side length $a$ be placed at origin with one vertex and one face along $x-y$ plane
let vertex at origin be $A(0,0,0)$
so opposite vertex will be $B(a,a,a)$
Let other vertex be $C(a,0,0)$
opposite vertex will be $D(0,a,a)$
Direction ratios of the two lines are $(a,a,a)$ and $(-a,a,a)$
$\cos \theta =\frac{-a\times a+a\times a+a\times a}{\sqrt{3}\times \sqrt{3}{a}^2}=\frac{1}{3}$