Consider a cube of side length a and whose adjacent edges are along positive x, y,z-axes. Which of the following are correct.

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The distance of diagonal of a cube not passing through origin from origin is $a \sqrt{\frac{2}{3}}$

Angles between diagonals of cube is $\cos^{- 1} (\frac{1}{3})$

The shortest distance between two faces diagonals of a cube which are skew is $\frac{a}{\sqrt{3}}$

Angles between diagonals of cube and diagonal of face of a cube $\cos^{- 1} (\frac{2}{3})$ is

answer is A, C, D.

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Detailed Solution

$Drs of \bar{O G} = ⟨ a, a, a ⟩$

$Drs of \bar{B D} = ⟨ a, - a, a ⟩$

$Dr s of \bar{G A} = ⟨ 0, a, a ⟩$

$Equation of Diagonal F A = \frac{x - a}{- a} = \frac{y - a}{a} = \frac{z - a}{a}$

Any point on FA

$(- a t, a + a t, a + a t) (O M ⊥ F A)$

$a^{2} t + a^{2} + d t + a^{2} + a^{2} t = 0$

$\Rightarrow 3 a^{2} t + 2 a^{2} = 0 t = \frac{- 2}{3}$

$M (\frac{2}{3} a, \frac{1}{3} a, \frac{1}{3} a) O M^{2} = \frac{4}{9} a^{2} + \frac{1}{9} a^{2} + \frac{1}{9} a^{2}$

$O M = \sqrt{\frac{2}{3} a}$

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