Question

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. $\frac{1}{\sqrt{3}}$

Answer 1

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

Vector along diagonal OD is $\overrightarrow{a}=(1-0)\hat{i}+(1-0)\hat{j}+(1-0)\hat{k}=\hat{i}+\hat{j}+\hat{k}$

And vector along diagonal AG is $\overrightarrow{b}=(0-1)\hat{i}+(1-0)\hat{j}+(1-0)\hat{k}=-\hat{i}+\hat{j}+\hat{k}$

If angle between the diagonal is $\theta$, then

$\cos \theta =\frac{\overrightarrow{a}\cdot \overrightarrow{b}}{|\overrightarrow{a}||\overrightarrow{b}|}=\frac{1(-1)+1\left(1\right)+1\left(1\right)}{\sqrt{3}\cdot \sqrt{3}}=\frac{1}{3}$