Question

The sum of the magnitudes of two vectors is 18. The magnitude of their resultant is 12. If the resultant is perpendicular to one of the vectors, then find the magnitudes of the two vectors.

Answer 1

$\text{Step 1: Draw the vectors}$$\text{[Ref. Fig.]}$

Consider there are two vectors $\overrightarrow{A}$ and $\overrightarrow{B}$ such that $\overrightarrow{B}$ is perpendicular to the resultant $\overrightarrow{R}$

Since the resultant is perpendicular to vector $\overrightarrow{B}$ we will get a right-angled triangle

$\text{Step 2:}$ $\text{Equation formation}$

From the triangle

$|\overrightarrow{A}{|}^2=|\overrightarrow{B}{|}^2+|\overrightarrow{R}{|}^2$ $....\left(1\right)$

Given , $|\overrightarrow{R}|=|\overrightarrow{A}+\overrightarrow{B}|=12$ $....\left(2\right)$

$|\overrightarrow{A}|+|\overrightarrow{B}|=18$ $....\left(3\right)$

$\text{Step 3:}$ $\text{Solving Equations}$

Equation $\left(3\right)$ and $\left(1\right)\Rightarrow$ $|\overrightarrow{A}{|}^2=(18-|A|{)}^2+(12{)}^2$

$\Rightarrow$ $|\overrightarrow{A}{|}^2={18}^2+|\overrightarrow{A}{|}^2-2\times 18|\overrightarrow{A}|+(12{)}^2$

$\Rightarrow$ $|\overrightarrow{A}|=13$

Equation $\left(3\right)\Rightarrow$ $|\overrightarrow{B}|=18-13=5$