Question

If $\overline{a}$ and $\overline{b}$ are unit vectors such that $\left[\begin{array}{ccc}\hat{a}& \hat{b}& \hat{a}\times \hat{b}\end{array}\right]$ is $\frac{1}{2}$, then the angle between $\overline{a}$ and $\overline{b}$ is

• A. $\frac{\pi }{3}$
• B. $\frac{\pi }{4}$
• C. $\frac{\pi }{6}$
• D. Data insufficient

Answer 1

Answer: (C)

$\frac{\pi }{6}$

We have $\left[\begin{array}{ccc}\overline{a}& \overline{b}& (\overline{a}\times \overline{b})\end{array}\right]=\frac{1}{4}$
$\Rightarrow$ $(\overline{a}\times \overline{b}).(\overline{a}\times \overline{b})=\frac{1}{4}$
$\Rightarrow$ ${|\overline{a}\times \overline{b}|}^2=\frac{1}{4}$
$\Rightarrow$ $\sin \theta =\frac{1}{2}$(as $\left|\overline{a}\right|\left|\overline{b}\right|=1$)
$\therefore$ $\theta =\frac{\pi }{6}$