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 $b$ and $c$, where b and c being non-parallel.

• A. a makes with b an angle ${60}^{\circ }$ and with c an angle of ${90}^{\circ }$.
• B. a makes with b an angle ${30}^{\circ }$ and with c an angle of ${90}^{\circ }$.
• C. a makes with b an angle ${90}^{\circ }$ and with c an angle of ${30}^{\circ }$.
• D. a makes with b an angle ${90}^{\circ }$ and with c an angle of ${60}^{\circ }$.

Answer 1

The correct option is D a makes with b an angle ${90}^{\circ }$ and with c an angle of ${60}^{\circ }$.

$(a.c)b-(a.b)c=\frac{1}{2}b$
or $(a.c-\frac{1}{2})b-(a.b)c=0$ or $xb-yc=0$
Above is a relation between b and c which are non-parallel
$\therefore x=0,y=0\therefore a.c=\frac{1}{2}$
or $1.1\cos \theta =\frac{1}{2}\therefore \theta ={60}^{\circ }$
and $a.b=0\therefore$ a and b are at right angles.
$\therefore$ a makes with b an angle ${90}^{\circ }$ and with c an angle of ${60}^{\circ }$.