Q. If $\overrightarrow{\mathbf{a}},\ \overrightarrow{\mathbf{b}},\ \overrightarrow{\mathbf{c}}$ be three unit vectors such that $\overrightarrow{\mathbf{a}} \times \left( \overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}} \right) = \frac{1}{2}\overrightarrow{\mathbf{b}},\ \overrightarrow{\mathbf{b}}\$ and $\ \overrightarrow{\mathbf{c}}$ being non-parallel. If $\theta_{1}$ is the angle between $\overrightarrow{\mathbf{a}}\ \ and\ \theta_{2}$ is the angle between $\overrightarrow{\mathbf{a}}$ and $\overrightarrow{\mathbf{c}}$, then
Explanation
(c)
$Since,\ \ \overrightarrow{\mathbf{a}} \times \left( \overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}} \right) = \frac{1}{2}\overrightarrow{\mathbf{b}}$
$\Longrightarrow \left( \overrightarrow{\mathbf{a}} \bullet \overrightarrow{\mathbf{b}} \right)\overrightarrow{\mathbf{b}} - \left( \overrightarrow{\mathbf{a}} \bullet \overrightarrow{\mathbf{b}} \right)\overrightarrow{\mathbf{c}} = \frac{1}{2}\overrightarrow{\mathbf{b}}$
On comparing both sides, we get
$\overrightarrow{\mathbf{a}} \bullet \overrightarrow{\mathbf{c}} = \frac{1}{2}\ and\ \overrightarrow{\mathbf{a}} \bullet \overrightarrow{\mathbf{b}} = 0$
$Now,\ \ \ \ \overrightarrow{\mathbf{a}} \bullet \overrightarrow{\mathbf{b}} = 0$
$\Longrightarrow \left| \overrightarrow{\mathbf{a}} \right|\left| \overrightarrow{\mathbf{c}} \right||\cos{\theta_{2}| = \frac{1}{2}\ \Longrightarrow \cos{\theta_{2} = \frac{1}{2} \Longrightarrow \theta_{2} = \frac{\pi}{3}}}$
and $\overrightarrow{\mathbf{a}} \bullet \overrightarrow{\mathbf{b}} = 0$
$\Longrightarrow \left| \overrightarrow{\mathbf{a}} \right|\left| \overrightarrow{\mathbf{b}} \right|\cos{\theta_{1} = 0}$
$\Longrightarrow \cos{\theta_{1} = \cos\frac{\pi}{2}}$
$\Longrightarrow \ \theta_{1} = \frac{\pi}{2}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.