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 $\overline{\mathrm{b}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}$. Let $\overline{\mathrm{c}}$ be a vector such that $|\bar{c}-\bar{a}|=3$ and $|(\overline{\mathrm{a}} \times \overline{\mathrm{b}}) \times \overline{\mathrm{c}}|=3$ 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. 2
  2. $-\frac{1}{8}$
  3. $\frac{25}{8}$
  4. 5

Solution

$\begin{array}{ll} & \bar{a}=2 \hat{i}+\hat{j}-2 \hat{k} \text { and } \bar{b}=\hat{i}+\hat{j} \\ & |a|=\sqrt{4+1+4}=3 \\ & \bar{a} \times \bar{b}=\left|\begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\ 2 & 1 & -2 \\ 1 & 1 & 0 \end{array}\right|=2 \hat{i}-2 \hat{j}+\hat{k} \\ \therefore \quad & |\bar{a} \times \bar{b}|=\sqrt{4+4+1}=3 \end{array}$
Angle between $\overline{\mathrm{c}}$ and $\overline{\mathrm{a}} \times \overline{\mathrm{b}}$ is $\frac{\pi}{6} \quad \ldots$ [Given] $\begin{aligned} \therefore \quad & \sin \frac{\pi}{6}=\frac{|(\overline{\mathrm{a}} \times \overline{\mathrm{b}}) \times \overline{\mathrm{c}}|}{|\overline{\mathrm{a}} \times \overline{\mathrm{c}}||\overline{\mathrm{c}}|} \\ & \Rightarrow \frac{1}{2}=\frac{3}{3 \times|\overline{\mathrm{c}}|} \\ \Rightarrow & |\overline{\mathrm{c}}|=2 \end{aligned}$
Now, $|\overline{\mathrm{c}}-\overline{\mathrm{a}}|=3$ $\begin{aligned} & \Rightarrow|\overline{\mathrm{c}}|^2+|\overline{\mathrm{a}}|^2-2 \overline{\mathrm{a}} \cdot \overline{\mathrm{c}}=9 \\ & \Rightarrow 4+9-2\mathrm{a} \cdot \mathrm{c}=9 \\ & \Rightarrow \mathrm{a} \cdot \mathrm{c}=2 \end{aligned}$

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

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