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} $
$\Rightarrow \sin ^{2} \theta=1-\frac{1}{9}=\frac{8}{9}$
$\therefore \sin \theta=\sqrt{\frac{8}{9}}=\frac{2 \sqrt{2}}{3}$