Question

Let $\overrightarrow{a}$ and $\overrightarrow{b}$ be two vectors such that $\left|\overrightarrow{a}\right|=3,\text{and}\left|\overrightarrow{b}\right|=\frac{\sqrt{2}}{3}.$ If $\overrightarrow{a}\times \overrightarrow{b}$ is a unit vector, then 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 $\frac{\pi }{4}$

Given:
$\left|\overrightarrow{a}\right|=3$ and $\left|\overrightarrow{b}\right|=\frac{\sqrt{2}}{3}$
We know that $\overrightarrow{a}\times \overrightarrow{b}=\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\sin \theta \hat{n}$
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 $|\overrightarrow{a}\times \overrightarrow{b}|=1$
$\Rightarrow \left|\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\sin \theta \hat{n}\right|=1$
$\Rightarrow \left|\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\sin \theta \right|=1$
$\Rightarrow 3\times \frac{\sqrt{2}}{3}\times \sin \theta =1$
$\Rightarrow \sin \theta =\frac{1}{\sqrt{2}}$
$\Rightarrow \theta =\frac{\pi }{4}$