Question

If $\overrightarrow{A}\times \overrightarrow{B}=\overrightarrow{B}\times \overrightarrow{A}$, then the angle between $A$ and $B$ is

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

Answer 1

The correct option is A $\pi$

$\text{Step 1: Simplify the given equation}$

$\overrightarrow{A}\times \overrightarrow{B}=\overrightarrow{B}\times \overrightarrow{A}$

$\Rightarrow \overrightarrow{A}\times \overrightarrow{B}-\overrightarrow{B}\times \overrightarrow{A}=0$

$\Rightarrow \overrightarrow{A}\times \overrightarrow{B}+\overrightarrow{A}\times \overrightarrow{B}=0$ $[\because \overrightarrow{A}\times \overrightarrow{B}=-\overrightarrow{B}\times \overrightarrow{A}]$

$\Rightarrow 2(\overrightarrow{A}\times \overrightarrow{B})=0$

$\Rightarrow 2\left|\overrightarrow{A}\right|\left|\overrightarrow{B}\right|\sin \theta =0$ $....\left(1\right)$

$\text{Step 2: Angle between A and B}$

Considering magnitudes of the vectors not to be zero.

Eq$\left(1\right)\Rightarrow$ $\sin \theta =0$

$\Rightarrow \theta =0\text{or}\pi$

$\because \theta =0$ is not in the options.

Hence, $\theta =\pi$, Option(A) is correct.