question_answer

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

Correct Answer: B

Solution :

Let the cube be of side\[a\]. Then, \[O(0,\,\,0,\,\,0),\,\,D(a,\,\,a,\,\,a),\,\,B(0,\,\,a,\,\,0)\]and\[G(a,\,\,0,\,\,a)\] Now, equations of diagonals \[OD\] 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                 \[{{\cos }^{-1}}\left( \frac{{{a}^{2}}-{{a}^{2}}+{{a}^{2}}}{\sqrt{3{{a}^{2}}}\cdot \sqrt{3{{a}^{2}}}} \right)={{\cos }^{-1}}\left( \frac{1}{3} \right)\]