Question

If $\overrightarrow{a}$ and $\overrightarrow{b}$ are perpendicular vectors, $|\overrightarrow{a}+\overrightarrow{b}|=13$ and $|\overrightarrow{a}|=5$, find the value of $|\overrightarrow{b}|$.

Answer 1

$As\overrightarrow{a}$ and $\overrightarrow{b}are\perp .$ $\therefore \overrightarrow{a}.\overrightarrow{b}=0...\left(i\right)$
$\Rightarrow |\overrightarrow{a}+\overrightarrow{b}|=13$

$\Rightarrow |\overrightarrow{a}|=5$

$\Rightarrow |\overrightarrow{a}+\overrightarrow{b}{|}^2={13}^2$

$\Rightarrow (\overrightarrow{a}+\overrightarrow{b}{)}^2=169,[\overrightarrow{|}a{|}^2=\overrightarrow{a}.\overrightarrow{a}$]

$\Rightarrow {\overrightarrow{a}}^2+{\overrightarrow{b}}^2+2.\overrightarrow{a}.\overrightarrow{b}=169,[|\overrightarrow{a}|=5,|\overrightarrow{a}{|}^2=5^2]$

$\Rightarrow (5{)}^2+{\overrightarrow{b}}^2+2\overrightarrow{a}.\overrightarrow{b}=169$

$[\overrightarrow{a}.\overrightarrow{b}=0]$

$\Rightarrow |\overrightarrow{b}{|}^2=169-25=144$

$\Rightarrow |\overrightarrow{b}|=12$