Question

If a, b, $c$ are non-zero vectors such that $(a\times b)\times c=\frac{1}{3}|b||c|a,c\perp a$ and $\theta$ is the angle between the vectors $b$ and $c$, then $\sin \theta$ is equal to

• A. $\frac{2\sqrt{2}}{3}$
• B. $\frac{1}{3}$
• C. $\frac{\sqrt{2}}{3}$
• D. $\frac{2}{3}$

Answer 1

Answer: (A)

Given, $(a\times b)\times c=\frac{1}{3}|b||c|a$
$(c\cdot a)b-(c\cdot b)a=\frac{1}{3}|b||c|a$
$\Rightarrow 0-(|c||b|\cos \theta )a=\frac{1}{3}|b||c|a[\because c\cdot a=0\Rightarrow c\perp a]$
$\Rightarrow \cos \theta =-\frac{1}{3}$
$\Rightarrow {\cos }^2\theta =\frac{1}{9}$
$\Rightarrow 1-{\sin }^2\theta =\frac{1}{9}$
$\Rightarrow {\sin }^2\theta =1-\frac{1}{9}=\frac{8}{9}$
$\therefore \sin \theta =\sqrt{\frac{8}{9}}=\frac{2\sqrt{2}}{3}$