Let the vectors a and b be such that |a|=3 and |b |= 2 3 , then a… - Vidyadip

MCQ

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

A
$\pi/6$
✓
$\pi/4$
C
$\pi/3$
D
$\pi/2$

✓

Answer

Correct option: B.

$\pi/4$

$\text{Given:}\ \ \big|\vec{a}\big|=3,\big|\vec{b}\big|=\frac{\sqrt{2}}{3}\ \text{and}\ \vec{a}\times\vec{b}$ is a unit vector.
$\Rightarrow\ \ \big|\vec{a}\times\vec{b}\big|=1\ $ $\Rightarrow\ \big|\vec{a}\big|.\Big|\vec{b}\Big|\ \text{sin}\ \theta=1,\ \text{where}\ \theta\ \text{is the angle between}\ \vec{a}\ \text{and}\ \vec{b}.$
$\Rightarrow\ \ 3\bigg(\frac{\sqrt{2}}{3}\bigg)\ \text{sin}\ \theta=1$ $\Rightarrow\ \sqrt{2}\ \text{sin}\ \theta=1\ \Rightarrow\ \ \text{sin}\ \theta=\frac{1}{\sqrt{2}}$
$\Rightarrow\ \text{sin}\ \theta=\text{sin}\frac{\pi}{{4}}\ \ \Rightarrow\ \theta=\frac{\pi}{4}$
Therefore, option (B) is correct.

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