Question

Let $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ be three non-zero vectors such that no two of them are collinear and $(\overrightarrow{a}\times\, \overrightarrow{b}) \times \overrightarrow{c}=\frac{1}{3}|\overrightarrow{b}||\overrightarrow{c}| \overrightarrow{a}.$ If $\theta$ is the angle between vectors $\overrightarrow{b}$ and $\overrightarrow{c}$ , then a value of sin $\theta$ is

• A. $\frac{2 \sqrt 2}{3}$
• B. $\frac{ -\sqrt 2}{3}$
• C. $\frac{2}{3}$
• D. $\frac{ -2 \sqrt 3}{3}$

Answer 1

Answer: (A)

Given, $ (\overrightarrow{a}\times\, \overrightarrow{b}) \times \overrightarrow{c}=\frac{1}{3}|\overrightarrow{b}||\overrightarrow{c}| \overrightarrow{a}.$
$\Rightarrow -\overrightarrow{c} \times (\overrightarrow{a}\times\, \overrightarrow{b}) =\frac{1}{3}|\overrightarrow{b}||\overrightarrow{c}| \overrightarrow{a}$
$\Rightarrow -(\overrightarrow{c}.\overrightarrow{b}). \overrightarrow{a}+ (\overrightarrow{c}. \overrightarrow{a}) \overrightarrow{b} =\frac{1}{3}|\overrightarrow{b}||\overrightarrow{c}| \overrightarrow{a}$
$\Rightarrow \bigg[\frac{1}{3} |\overrightarrow{b}||\overrightarrow{c}|+(\overrightarrow{c}.\overrightarrow{b})\bigg] a = (\overrightarrow{c}. \overrightarrow{a}) \overrightarrow{b} $
Since, $a$ and $b$ are not collinear.
$\therefore \overrightarrow{c}.\overrightarrow{b}+ \frac{1}{3} |\overrightarrow{b}||\overrightarrow{c}|= 0\,$ and $\, \overrightarrow{c}. \overrightarrow{a} = 0 $
$\Rightarrow |\overrightarrow{b}||\overrightarrow{c}|\, \cos\theta + \frac{1}{3} |\overrightarrow{b}||\overrightarrow{c}|= 0 $
$\Rightarrow |\overrightarrow{b}||\overrightarrow{c}|\bigg( \cos\theta + \frac{1}{3} \bigg) = 0 $
$\Rightarrow \cos\theta + \frac{1}{3} = 0 [\because\, |\, b\, | \ne 0 , |\, c \, | \ne 0]$
$\Rightarrow \cos\theta =- \frac{1}{3} \Rightarrow \sin\theta = \frac{\sqrt 8}{3} = \frac{2\sqrt 2}{3} $