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

Let $\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}$ be three vectors such that $|\overline{\mathrm{a}}|=\sqrt{3}$, $|\bar{b}|=5, \bar{b} \cdot \bar{c}=10$ and the angle between $\bar{b}$ and $\bar{c}$ is $\frac{\pi}{3}$. If $\overline{\mathrm{a}}$ is perpendicular to the vector $\overline{\mathrm{b}} \times \overline{\mathrm{c}}$, then $|\overline{\mathrm{a}} \times(\overline{\mathrm{b}} \times \overline{\mathrm{c}})|$ is equal to
  1. $10 \sqrt{3}$
  2. $5 \sqrt{3}$
  3. $60$
  4. $30$

Solution

$\begin{aligned} & \overline{\mathrm{b}} \cdot \overline{\mathrm{c}}=10 \\ & \Rightarrow|\overline{\mathrm{b}}||\overline{\mathrm{c}}| \cos \frac{\pi}{3}=10 \\ & \Rightarrow(5)|\overline{\mathrm{c}}|\left(\frac{1}{2}\right)=10 \\ & \Rightarrow|\overline{\mathrm{c}}|=4\end{aligned}$ $\begin{aligned} & |\overline{\mathrm{a}} \times(\overline{\mathrm{b}} \times \overline{\mathrm{c}})|=|\overline{\mathrm{a}}||\overline{\mathrm{b}} \times \overline{\mathrm{c}}| \sin \frac{\pi}{2} \\ & =|\overline{\mathrm{a}}||\overline{\mathrm{b}} \times \overline{\mathrm{c}}| \\ & =|\overline{\mathrm{a}}||\overline{\mathrm{b}}||\overline{\mathrm{c}}| \sin \frac{\pi}{3} \\ & =(\sqrt{3})(5)(4)\left(\frac{\sqrt{3}}{2}\right) \\ & =30\end{aligned}$

Asked in: MHT CET 2023 (14 May Shift 1)

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