Question

What is the angle between $\overrightarrow{P}$ and the resultant of $(\overrightarrow{P}+\overrightarrow{Q})$ and $(\overrightarrow{P}-\overrightarrow{Q})$?

• A. $\frac{{\tan }^{-1}|(\overrightarrow{P}-\overrightarrow{Q})|}{|P+Q|}$
• B. ${\tan }^{-1}\left(Q/P\right)$
• C. ${\tan }^{-1}\left(P/Q\right)$
• D. zero

Answer 1

The correct option is D zero

$\text{Step 1: Resultant Calculation}$

Let the resultant be $\overrightarrow{R}$

$\therefore \overrightarrow{R}=(\overrightarrow{P}+\overrightarrow{Q})+(\overrightarrow{P}-\overrightarrow{Q})$

$\Rightarrow$ $\overrightarrow{R}=2\overrightarrow{P}$ $....\left(1\right)$

$\text{Step 2: Angle between}\overrightarrow{R}\text{and}\overrightarrow{P}$

From $\left(1\right)$ we can say that, $\overrightarrow{R}$ is in same direction along $\overrightarrow{P}$ with twice its magnitude. Thus,$\overrightarrow{R}$ and $\overrightarrow{P}$ are parallel.

Therefore, the angle between $\overrightarrow{R}$ and $\overrightarrow{P}$ is zero.

Hence, Option $D$ is correct