Let $\overrightarrow{\mathrm{a}}=6 \hat{i}+\hat{j}-\hat{k}$ and…

Let $\overrightarrow{\mathrm{a}}=6 \hat{i}+\hat{j}-\hat{k}$ and $\overrightarrow{\mathrm{b}}=\hat{i}+\hat{j}$. If $\overrightarrow{\mathrm{c}}$ is a is vector such that $|\overrightarrow{\mathrm{c}}| \geq 6, \overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{c}}=6|\overrightarrow{\mathrm{c}}|,|\overrightarrow{\mathrm{c}}-\overrightarrow{\mathrm{a}}|=2 \sqrt{2}$ and the angle between $\vec{a} \times \vec{b}$ and $\vec{c}$ is $60^{\circ}$, then $|(\vec{a} \times \vec{b}) \times \vec{c}|$ is equal to:
  1. $\frac{9}{2}(6-\sqrt{6})$
  2. $\frac{3}{2} \sqrt{6}$
  3. $\frac{9}{2}(6+\sqrt{6})$
  4. $\frac{3}{2} \sqrt{3}$

Solution

$\begin{aligned} & |(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}})|=|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}||\overrightarrow{\mathrm{c}}| \frac{\sqrt{3}}{2} \\ & |\overrightarrow{\mathrm{c}}-\overrightarrow{\mathrm{a}}|=2 \sqrt{2} \\ & |\mathrm{c}|^2+|\mathrm{a}|^2-2 \overrightarrow{\mathrm{c}} \cdot \overrightarrow{\mathrm{a}}=8 \\ & |\mathrm{z}|^2+38-12|\mathrm{z}|=8 \\ & |\mathrm{z}|^2-12|\mathrm{z}|+30=0 \\ & |\mathrm{z}|=\frac{12 \pm \sqrt{144-120}}{2} \\ & =\frac{12 \pm 2 \sqrt{6}}{2}\end{aligned}$ $\begin{aligned} & |z|=6+\sqrt{6} \\ & \overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}=\left|\begin{array}{ccc}\hat{\ell} & \hat{\mathrm{j}} & \hat{\mathrm{k}} \\ 6 & 1 & -1 \\ 1 & 1 & 0\end{array}\right| \\ & \hat{\ell}-\hat{\mathrm{j}}+5 \hat{\mathrm{k}} \\ & |\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}|=\sqrt{27} \\ & |(\overrightarrow{\mathrm{a}} \times \mathrm{b}) \times \mathrm{z}|=\sqrt{27}(6+\sqrt{6}) \frac{\sqrt{3}}{2} \\ & \frac{9}{2}(6+\sqrt{6})\end{aligned}$

Asked in: JEE Main 2024 (06 Apr Shift 2)

Practice more Vectors questions on Aicharya