Question

Let the vectors a and b be such that $|a|=3$ and $|b|=\frac{\sqrt{2}}{3}$, then $a\times b$ is a unit vector, if the angle between a and b is

a) $\frac{\pi }{6}$

b) $\frac{\pi }{4}$

c) $\frac{\pi }{3}$

d) $\frac{\pi }{2}$

Answer 1

It is given that $|a|=3$ and $|b|=\frac{\sqrt{2}}{3}$
Let $\theta$ be the angle between a and b, then $|a\times b|=1$
$\Rightarrow |a||b|sin\theta =1\Rightarrow 3\times \frac{\sqrt{2}}{3}\times sin\theta =1\Rightarrow sin\theta =\frac{1}{\sqrt{2}}\Rightarrow \theta =\frac{\pi }{4}$
Hence, $a\times b$ is a unit vector, if the angle between a and b is $\frac{\pi }{4}$.