Question

If a,b,c be three unit vectors such that $a\times (b\times c)=\frac{1}{2}b$, find the angles which a makes with c, b and c being non-parallel.

• A. $\frac{\pi }{3}$
• B. $\frac{\pi }{6}$
• C. $\frac{2\pi }{3}$
• D. None of these

Answer 1

The correct option is A $\frac{\pi }{3}$

$a\times (b\times c)=\frac{1}{2}b$

It can be written as

$(a.c)b-(a.b)c=\frac{1}{2}b$

$\left((a.c)-\frac{1}{2}\right)b-(a.b)c=0$

$a.c-\frac{1}{2}=0$

Since, $b$ and $c$ are't parallel, so coefficient of $b$ are $0$

Therefore, $a.c=\frac{1}{2}$

$|a||c|\cos \theta =\frac{1}{2}$

where $\theta$ is angle between the vectors $a$ and $C$

and $a$ and $c$ are unit vectors.

therefore,

$1\times 1\times \cos \theta =\frac{1}{2}$

$\cos \theta =\frac{1}{2}$

Hence, $\theta ={60}^{\circ }$

Therefore angle that makes $a,b,c$ non-parallel is $\frac{\pi }{3}$