Question

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

• A. $90^{\circ },60^{\circ }$
• B. $45^{\circ },60^{\circ }$
• C. $30^{\circ },60^{\circ }$
• D. none of these

Answer 1

Answer: (A)

We have, $a\times (b\times c)=\frac{1}{2}b$
$\therefore (a\cdot c)b-(a\cdot b)c=\frac{1}{2}b$
$\Rightarrow \left(a\cdot c-\frac{1}{2}\right)b-(a\cdot b)c=0(1)$
Since $b$ and $c$ are non-parallel , (Given)
$\therefore$ (i) exists if coefficients of $b$ and $c$ vanish separately
i.e., $a\cdot c-\frac{1}{2}=0$ and $a\cdot b=0$.
$\Rightarrow a\cdot c=\frac{1}{2}$ and $a\cdot b=0$
Let ${\theta }_1$ and ${\theta }_2$ be the angles which $a$ makes with $b$ and $c$, respectively.
Also $a,b,c$ are unit vectors,
$\therefore \cos {\theta }_2=\frac{1}{2}$ and $\cos {\theta }_1=0$
i.e., ${\theta }_2=60^{\circ }$ and ${\theta }_1=90^{\circ }$
Hence, ${\theta }_1=90^{\circ }$ and ${\theta }_2=60^{\circ }$.