Question

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

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

Answer 1

Answer: (B)

$\frac{\pi}{4}$

Answer 2

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