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

Answer 1

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

$a\times (b\times c)=\frac{b+c}{\sqrt{2}}\Rightarrow (a.c)b-(a.b)c=\frac{b+c}{\sqrt{2}}\Rightarrow \left[\right(a.c)-\frac{1}{\sqrt{2}}]b-\left[\right(a.b)+\frac{1}{\sqrt{2}}]c=0\Rightarrow a.c=\frac{1}{\sqrt{2}},a.b=-\frac{1}{\sqrt{2}}$
$\Rightarrow |a||c|cos\theta =\frac{1}{\sqrt{2}},|a||b|cos\varphi =-\frac{1}{\sqrt{2}}\Rightarrow cos\theta =\frac{1}{\sqrt{2}},cos=-\frac{1}{\sqrt{2}}\Rightarrow \theta =\frac{\pi }{4},\varphi =\frac{3\pi }{4}.$