$\hat{a}, \hat{b}$, and $\hat{c}$ are three unit vectors such that $\hat{a} \times(\hat{b} \times…

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

Solution

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

Asked in: MHT CET 2024 (02 May Shift 1)

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