Question

The sum of two forces $\overrightarrow{P}$ and $\overrightarrow{Q}$​ is $\overrightarrow{R}$ such that $\left|\overrightarrow{R}\right|=\left|\overrightarrow{P}\right|$. The angle $\theta$ (in degrees) that the resultant of $2\overrightarrow{P}$ and $\overrightarrow{Q}$​ will make with $\overrightarrow{Q}$​ is, ______.

Answer 1

Step 1: Given Data

Sum of the two forces, $\overrightarrow{P}+\overrightarrow{Q}=\overrightarrow{R}$

$\left|\overrightarrow{R}\right|=\left|\overrightarrow{P}\right|$

the angle between resultant of $2\overrightarrow{P}+\overrightarrow{Q}$ and $\overrightarrow{Q}$​ is $\theta$

the angle between resultant of $\overrightarrow{R}$ and $\overrightarrow{Q}$​ is $\varphi$

Step 2: Formula Used

Sum of the angles of a triangle$=90°$

Step 3: Calculate the value of $\theta$

Let $\angle BOA=\varphi$ +
ln $\Delta OBE$

$BE=OB$

$\angle BEO=\angle BOE=\theta -\varphi$

ln $\Delta OBE$

$\angle BEO+\angle BOE+\angle EBO={180}^{\circ }$

$\Rightarrow (\theta -\varphi )+2\varphi +(\theta -\varphi )={180}^{\circ }$

$\Rightarrow 2\theta +2\varphi -2\varphi ={180}^{\circ }$

$\Rightarrow 2\theta ={180}^{\circ }$

$\Rightarrow \theta ={90}^{\circ }$

So, vector $2\overrightarrow{P}+\overrightarrow{Q}$ will make ${90}^{\circ }$ with $\overrightarrow{Q}$​.

Hence, the angle $\theta$ (in degrees) that the resultant of $2\overrightarrow{P}$ and $\overrightarrow{Q}$​ will make with $\overrightarrow{Q}$​ is, $90°$.