Question

If the resultant of two forces is of magnitude $P$ and equal to one of them and perpendicular to it, then the magnitude of other force is

• A. $\sqrt{2}P$
• B. $P$
• C. $\sqrt{3}P$
• D. $2\sqrt{2}P$

Answer 1

The correct option is A $\sqrt{2}P$

Let, $\overrightarrow{R}=\overrightarrow{P}+\overrightarrow{Q}$
Where $|\overrightarrow{R}|=|\overrightarrow{P}|=P$, also
$\overrightarrow{R}$ is perpendicular to $\overrightarrow{P}$
$\Rightarrow \overrightarrow{R}\cdot \overrightarrow{P}=0$
$\Rightarrow (\overrightarrow{P}+\overrightarrow{Q})\cdot \overrightarrow{P}=0$
$\Rightarrow P^2+\overrightarrow{Q}\cdot \overrightarrow{P}=0\cdots \left(1\right)$
$R^2=Q^2+P^2+2\overrightarrow{Q}\cdot \overrightarrow{P}=P^2$
$\Rightarrow Q^2+2\overrightarrow{Q}\cdot \overrightarrow{P}=0\cdots \left(2\right)$
From $\left(1\right)$ and $\left(2\right)$, we get
$Q^2=2P^2$
$\Rightarrow |\overrightarrow{Q}|=\sqrt{2}P$