$\mathbf{a}, \mathbf{b}$ are two vectors such that $|\mathbf{a}|=\sqrt{3},|\mathbf{b}|=\sqrt{2}$. If…

$\mathbf{a}, \mathbf{b}$ are two vectors such that $|\mathbf{a}|=\sqrt{3},|\mathbf{b}|=\sqrt{2}$. If $\mathbf{x}$ is unit vector satisfying $\mathbf{x} \times \mathbf{a}=\mathbf{b}$, then $\mathbf{x}=$
  1. $\frac{1}{2}[(x \cdot a) a-b \times a]$
  2. $\frac{1}{2}[ \pm(x \cdot a) a+(b \times a)]$
  3. $\frac{1}{2}[(x \cdot a) a+b \times a]$
  4. $-8 \hat{i}-12 \hat{i}+24 \hat{k}$

Solution

$\begin{aligned} & \mathbf{a} \times(\mathbf{x} \times \mathbf{a})=\mathbf{a} \times \mathbf{b} \\ \Rightarrow & (\mathbf{a} \cdot \mathbf{a}) \mathbf{x}-(\mathbf{a} \cdot \mathbf{x}) \mathbf{a}=\mathbf{a} \times \mathbf{b} \quad\left[\therefore|\mathbf{a}|=\sqrt{3},|\mathbf{a}|^2=3\right] \\ \Rightarrow & 3 \mathbf{x}=\mathbf{a} \times \mathbf{b}+(\mathbf{a} \cdot \mathbf{x}) \mathbf{a} \\ \Rightarrow & 3 \mathbf{x}=\mathbf{a} \times \mathbf{b} \pm\left(\sqrt{3} \times \frac{1}{\sqrt{3}}\right) \mathbf{a} \\ \Rightarrow & \mathbf{x}=\frac{1}{3}(\mathbf{a} \times \mathbf{b} \pm \mathbf{a}) \\ & |\mathbf{a} \times \mathbf{x}|=|\mathbf{b}| \\ \Rightarrow & \sqrt{3} \times 1 \times \sin \theta=\sqrt{2} \\ \Rightarrow & \quad \sin \theta=\sqrt{\frac{2}{3}} \\ \Rightarrow & \quad \cos \theta= \pm \frac{1}{\sqrt{3}}\end{aligned}$

Asked in: AP EAMCET 2022 (07 Jul Shift 2)

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