Question

If $\overrightarrow{a}$ and $\overrightarrow{b}$ are perpendicular vectors, $\left|\overrightarrow{a}+\overrightarrow{b}\right|=3$ and $\left|\overrightarrow{a}\right|=5$, find the value of $\left|\overrightarrow{b}\right|$. [CBSE 2014]

Answer 1

Disclaimer: $\left|\overrightarrow{a}+\overrightarrow{b}\right|=13$ has been taken in order to solve the question.

It is given that $\overrightarrow{a}$ and $\overrightarrow{b}$ are perpendicular vectors.

$\therefore \overrightarrow{a}.\overrightarrow{b}=0$ .....(1)

$$\begin{gathered} \left|\overrightarrow{a}+\overrightarrow{b}\right|=13 \\ \Rightarrow {\left|\overrightarrow{a}+\overrightarrow{b}\right|}^2=169 \\ \Rightarrow {\left|\overrightarrow{a}\right|}^2+2\overrightarrow{a}.\overrightarrow{b}+{\left|\overrightarrow{b}\right|}^2=169 \\ \Rightarrow 25+2\times 0+{\left|\overrightarrow{b}\right|}^2=169\left[\mathrm{Using}\left(1\right)\right] \end{gathered}$$
$$\begin{gathered} \Rightarrow {\left|\overrightarrow{b}\right|}^2=169-25=144 \\ \Rightarrow \left|\overrightarrow{b}\right|=12 \end{gathered}$$

Thus, the value of $\left|\overrightarrow{b}\right|$ is 12.