Question

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

• A. $\frac{\pi }{6}$
• B. $\frac{\pi }{4}$
• C. $\frac{\pi }{3}$
• D. $\frac{\pi }{2}$

Answer 1

The correct option is B $\frac{\pi }{4}$

It is given that $|\overrightarrow{a}|=3$ and $|\overrightarrow{b}|=\frac{\sqrt{2}}{3}$.
We know that $\overrightarrow{a}\times \overrightarrow{b}=|\overrightarrow{a}||\overrightarrow{b}|\sin \theta \hat{n}$, where $\hat{n}$ is a unit vector perpendicular to both $\overrightarrow{a}$ and $\overrightarrow{b}$ and $\theta$ is the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$.
Now, $\overrightarrow{a}\times \overrightarrow{b}$ is a unit vector if $|\overrightarrow{a}\times \overrightarrow{b}|=1$
$|\overrightarrow{a}\times \overrightarrow{b}|=1$
$\Rightarrow \left||\overrightarrow{a}||\overrightarrow{b}|\sin \theta \hat{n}\right|=1$
$\Rightarrow |\overrightarrow{a}||\overrightarrow{b}||\sin \theta |=1$
$\Rightarrow 3\times \frac{\sqrt{2}}{3}\times \sin \theta =1$
$\Rightarrow \sin \theta =\frac{1}{\sqrt{2}}$
$\Rightarrow \theta =\frac{\pi }{4}$
Hence, $\overrightarrow{a}\times \overrightarrow{b}$ is a unit vector if the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\frac{\pi }{4}$.
The correct answer is $B$.