Let $\overline{\mathrm{a}}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-2 \hat{\mathrm{k}}$ and…

Let $\overline{\mathrm{a}}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-2 \hat{\mathrm{k}}$ and $\overline{\mathrm{b}}=\hat{i}+\hat{\mathbf{j}}$. Let $\overline{\mathrm{c}}$ be a vector such that $|\bar{c}-\bar{a}|=3$ and $|(\bar{a} \times \bar{b}) \times \bar{c}|=3$ and the angle between $\overline{\mathrm{c}}$ and $\overline{\mathrm{a}} \times \overline{\mathrm{b}}$ is $30^{\circ}$, then $\overline{\mathrm{a}} \cdot \overline{\mathrm{c}}$ is equal to
  1. $\frac{2 \sqrt{2}}{3}$
  2. 5
  3. $-\frac{1}{8}$
  4. 2

Solution

$\begin{array}{ll} & \bar{a}=2 \hat{i}+\hat{j}-2 \hat{k} \\ & \bar{b}=\hat{i}+\hat{j} \\ & |\bar{a}|=\sqrt{4+1+4}=3 \\ & \bar{a} \times \bar{b}=\left|\begin{array}{ccc}\hat{i} & \hat{j} & \hat{k} \\ 2 & 1 & -2 \\ 1 & 1 & 0\end{array}\right|=2 \hat{i}-2 \hat{j}+\hat{k} \\ \therefore \quad & |\bar{a} \times \bar{b}|=\sqrt{4+4+1}=3\end{array}$ Angle between $\overline{\mathrm{c}}$ and $\overline{\mathrm{a}} \times \overline{\mathrm{b}}=30^{\circ}$ ...[Given] $\sin 30^{\circ}=\frac{|(\overline{\mathrm{a}} \times \overline{\mathrm{b}}) \times \overline{\mathrm{c}}|}{|\overline{\mathrm{a}} \times \overline{\mathrm{b}}||\overline{\mathrm{c}}|}$ $\begin{aligned} & \frac{1}{2}=\frac{3}{3 \times|\overline{\mathrm{c}}|} \\ & \Rightarrow|\overline{\mathrm{c}}|=2 \end{aligned}$ $\begin{aligned} & \text { Now, }|\overline{\mathrm{c}}-\overline{\mathrm{a}}|=3 \\ & \Rightarrow|\overline{\mathrm{c}}|^2+|\overline{\mathrm{a}}|^2-2 \overline{\mathrm{a}} \cdot \overline{\mathrm{c}}=9 \\ & \Rightarrow 4+9-2 \overline{\mathrm{a}} \cdot \overline{\mathrm{c}}=9 \\ & \Rightarrow \overline{\mathrm{a}} \cdot \overline{\mathrm{c}}=2 \end{aligned}$

Asked in: MHT CET 2024 (04 May Shift 2)

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