Question

If $\overrightarrow{a}$ and $\overrightarrow{b}$ are unit vectors such that $[\overrightarrow{a}\overrightarrow{b}\overrightarrow{a}\times \overrightarrow{b}]=\frac{1}{4},$ then angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is:

• A. $\frac{\pi }{3}$
• B. $\frac{\pi }{4}$
• C. $\frac{\pi }{6}$
• D. $\frac{\pi }{2}$
• E. $\frac{\pi }{12}$

Answer 1

Answer: (C)

$|\overrightarrow{a}|=1,|\overrightarrow{b}|=1,[\overrightarrow{a},\overrightarrow{b},\overrightarrow{a}\times \overrightarrow{b}]=\frac{1}{4}$ $\Rightarrow$ $(\overrightarrow{a}\times \overrightarrow{b}).(\overrightarrow{a}\times \overrightarrow{b})=\frac{1}{4}$ $\Rightarrow$ ${(|\overrightarrow{a}||\overrightarrow{b}|\sin \theta )}^2=\frac{1}{4}\Rightarrow {\sin }^2\theta =\frac{1}{4}$ $\Rightarrow$ $\sin \theta =\frac{1}{2}$ $\Rightarrow$ $\theta =\frac{\pi }{6}$