Question

$a,b,c$ are three unit vectors such that $a\times (b\times c)=\frac{1}{2}(b+c)$. If the vectors $b$ and $c$ are non-parallel, then the angle between $a$ and $b$ is equal to

• A. $\frac{\pi }{3}$
• B. $\frac{2\pi }{3}$
• C. $\frac{5\pi }{6}$
• D. none of these

Answer 1

The correct option is B $\frac{2\pi }{3}$

Let angles between $a$ and $b$ and between $a$ and $c$ br $\alpha$ and $\beta$ respectively.

We have, $a\times (b\times c)=\frac{1}{2}(b+c)$

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

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

$\Rightarrow a.c=\frac{1}{2},a.b=-\frac{1}{2}$ $($ As $b$ and $c$ are non parallel $)$

$\Rightarrow \cos \beta =\frac{1}{2},\cos \alpha =-\frac{1}{2}[a,b,c$ are unit vectors $]$

$\Rightarrow \beta =\frac{\pi }{3},\alpha =\frac{2\pi }{3}$

Hence angle between $a$ and $b$ is $\frac{2\pi }{3}$