Question

If $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ are three coplanar unit vectors such that $\overrightarrow{a}\times (\overrightarrow{b}\times \overrightarrow{c})=\frac{\overrightarrow{b}}{2}$, then the angle between $\overrightarrow{b}$ and $\overrightarrow{c}$ can be

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

Answer 1

The correct option is B $\frac{\pi }{6}$

Given that $\overrightarrow{a}\times (\overrightarrow{b}\times \overrightarrow{c})=\frac{\overrightarrow{b}}{2}$

$\Rightarrow (\overrightarrow{a}.\overrightarrow{c})\overrightarrow{b}-(\overrightarrow{a}.\overrightarrow{b})\overrightarrow{c}=\frac{\overrightarrow{b}}{2}\Rightarrow \overrightarrow{a}.\overrightarrow{c}=\frac{1}{2}$ and $\overrightarrow{a}.\overrightarrow{b}=0$

Hence angle between $\overrightarrow{a}$ and $\overrightarrow{c}$ is $\frac{\pi }{3}$ and angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\frac{\pi }{2}$

$\therefore$ Angle between $\overrightarrow{b}$ and $\overrightarrow{c}$ can be $(\frac{\pi }{2}-\frac{\pi }{3})=\frac{\pi }{6}$ or $(\frac{\pi }{2}+\frac{\pi }{3})=\frac{5\pi }{6}$