Question

If $\overrightarrow{P}\times \overrightarrow{Q}=\overrightarrow{Q}\times \overrightarrow{P}$, the angle between $\overrightarrow{P}$ and $\overrightarrow{Q}$ is $\theta (0°<\theta <360°)$, The value of $\theta$ will be

Answer 1

Step 1: Given data:

$\overrightarrow{P}\times \overrightarrow{Q}=\overrightarrow{Q}\times \overrightarrow{P}$

Angle between $\overrightarrow{P}$ and $\overrightarrow{Q}$ $=\theta (0°<\theta <360°)$

$\theta =?$

Step 2: Calculation of $\theta$

$\overrightarrow{P}\times \overrightarrow{Q}=\overrightarrow{Q}\times \overrightarrow{P}$

$\overrightarrow{P}\times \overrightarrow{Q}=-\overrightarrow{P}\times \overrightarrow{Q}$ $[\overrightarrow{A}\times \overrightarrow{B}=-\overrightarrow{A}\times \overrightarrow{B}]$

$\overrightarrow{P}\times \overrightarrow{Q}+\hspace{0.17em}\overrightarrow{P}\times \overrightarrow{Q}=0$

$2(\overrightarrow{P}\times \overrightarrow{Q})=0$

or $\overrightarrow{P}\times \overrightarrow{Q}=0$

But this is simply possible if $\overrightarrow{P}=0$ or $\overrightarrow{Q}=0$ or the angle between $\overrightarrow{P}$ and $\overrightarrow{Q}$ = $180°(0°<\theta <360°)$

So, $\theta =180°$

Hence, the value of $\theta$ will be $180°$.