Question

If $a,b,c$ are non-coplanar unit vectors such that $a\times (b\times c)=\frac{b+c}{\sqrt{2}},$ then the angle between $a$ and $b$ is

• A. $\frac{3\pi }{4}$
• B. $\frac{\pi }{4}$
• C. $\frac{\pi }{2}$
• D. $\pi$

Answer 1

The correct option is A $\frac{3\pi }{4}$

$\frac{b+c}{\sqrt{2}}=a\times (b\times c)=(a.c)b-(a.b)c$

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

Since $a,b,c$ are non-coplanar so $a,b,c$ are linearly independent.

Hence, $a.b=-\frac{1}{\sqrt{2}}$

$\therefore \cos \theta =\frac{a.b}{\left|a\right|\left|b\right|}=a.b=-\frac{1}{\sqrt{2}}$ $(\theta$ is the angle between $a$ and $b)$

$\Rightarrow \theta =\frac{3\pi }{4}$