Let $\overline{\mathrm{a}}, \overline{\mathrm{b}}$ and $\overline{\mathrm{c}}$ be non-zero vectors such that…

Let $\overline{\mathrm{a}}, \overline{\mathrm{b}}$ and $\overline{\mathrm{c}}$ be non-zero vectors such that $(\overline{\mathrm{a}} \times \overline{\mathrm{b}}) \times \overline{\mathrm{c}}=\frac{1}{3}|\overline{\mathrm{b}}||\overline{\mathrm{c}}| \overline{\mathrm{a}}$. If $\theta$ is the acute angle between the vectors $\overline{\mathrm{b}}$ and $\overline{\mathrm{c}}$, then $\sin \theta$ equals
  1. $\frac{1}{3}$
  2. $\frac{\sqrt{2}}{3}$
  3. $\frac{2}{3}$
  4. $\frac{2 \sqrt{2}}{3}$

Solution

$(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}) \times \overrightarrow{\mathrm{c}}=\frac{1}{3}|\overrightarrow{\mathrm{b}}||\overline{\mathrm{c}}| \overline{\mathrm{a}} \Rightarrow(\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{c}}) \overrightarrow{\mathrm{b}}-(\overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{c}}) \overrightarrow{\mathrm{a}}=\frac{1}{3}|\overline{\mathrm{b}}||\overline{\mathrm{c}}| \overline{\mathrm{a}}$ $\Rightarrow(\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{c}}) \overrightarrow{\mathrm{b}}=\left(\frac{1}{3}|\overline{\mathrm{b}}||\overline{\mathrm{c}}|+(\overline{\mathrm{b}} \cdot \overline{\mathrm{c}})\right) \overrightarrow{\mathrm{a}} \Rightarrow \overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{c}}=0$ and $\frac{1}{3}|\overline{\mathrm{b}}||\overline{\mathrm{c}}|+(\overline{\mathrm{b}} \cdot \overline{\mathrm{c}})=0$ $\Rightarrow|\overline{\mathrm{b}}||\overline{\mathrm{c}}|\left(\frac{1}{3}+\cos \theta\right)=0 \Rightarrow \cos \theta=-1 / 3 \Rightarrow \sin \theta=\frac{2 \sqrt{2}}{3}$

Asked in: JEE Main 2004

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