question_answer

The 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)  \[\frac{1}{\sqrt{3}}\]

Correct Answer: A

Solution :

 Let every side of a cube is a and O, A, G, D are its vertex \[\therefore \]Direction cosines of OG and AD are \[\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}\]and\[\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}}\] Cosine of the angle between OG and AD ie, \[\cos \theta =ll'+mm'+nn'\] \[=\frac{1}{3}+\frac{1}{3}-\frac{1}{3}=\frac{1}{3}\]