Let $\overline{\mathrm{a}}, \overline{\mathrm{b}}$ and $\overline{\mathrm{c}}$ be three unit vectors such…

Let $\overline{\mathrm{a}}, \overline{\mathrm{b}}$ and $\overline{\mathrm{c}}$ be three unit vectors such that $\overline{\mathrm{a}} \times(\overline{\mathrm{b}} \times \overline{\mathrm{c}})=\frac{\sqrt{3}}{2}(\overline{\mathrm{b}}+\overline{\mathrm{c}})$. If $\overline{\mathrm{b}}$ is not parallel to $\overline{\mathrm{c}}$, then the angle between $\bar{a}$ and $\bar{b}$ is
  1. $\frac{5 \pi}{6}$
  2. $\frac{2 \pi}{3}$
  3. $\frac{\pi}{6}$
  4. $\frac{\pi}{3}$

Solution

$\begin{aligned} & \overline{\mathrm{a}} \times(\overline{\mathrm{b}} \times \overline{\mathrm{c}})=\frac{\sqrt{3}}{2}(\overline{\mathrm{b}}+\overline{\mathrm{c}}) \\ & \Rightarrow(\overline{\mathrm{a}} \cdot \overline{\mathrm{c}}) \overline{\mathrm{b}}-(\overline{\mathrm{a}} \cdot \overline{\mathrm{b}}) \overline{\mathrm{c}}=\frac{\sqrt{3}}{2} \overline{\mathrm{b}}+\frac{\sqrt{3}}{2} \overline{\mathrm{c}} \end{aligned}$ On comparing, we get $\begin{aligned} & \overline{\mathrm{a}} \cdot \overline{\mathrm{c}}=\frac{\sqrt{3}}{2} \text { and } \overline{\mathrm{a}} \cdot \overline{\mathrm{b}}=-\frac{\sqrt{3}}{2} \\ & \Rightarrow|\overline{\mathrm{a}}||\overline{\mathrm{b}}| \cos \theta=-\frac{\sqrt{3}}{2} \\ & \Rightarrow \cos \theta=-\frac{\sqrt{3}}{2} \\ & \Rightarrow \theta=\frac{5 \pi}{6} \end{aligned}$

Asked in: MHT CET 2023 (11 May Shift 1)

Practice more Vector Algebra questions on Aicharya