Question

If $a,b,c$ are three unit vectors such that $a\times (b\times c)=\frac{1}{2}b$ and $c$ being non parallel, then

• A. angle between $a$ and $b$ is $\pi /2$
• B. angle between $a$ and $c$ is $\pi /4$
• C. angle between $a$ and $c$ is $\pi /3$
• D. angle between $a$ and $b$ is $\pi /3$

Answer 1

Answer: (A), (C)

The correct options are
A angle between $a$ and $b$ is $\pi /2$
C angle between $a$ and $c$ is $\pi /3$

$a\times (b\times c)=(a\cdot c)b-(a\cdot b)c=\frac{1}{2}b$
$\Rightarrow (a\cdot c-1/2)b=(a\cdot b)c$
But $b$ and $c$ are non parallel so
$a\cdot c-1/2=0,a\cdot b=0$
$\Rightarrow$ angle between $a$ and $c$ is $\pi /3$ and the angle between $a$ and $b$ is $\pi /2$