# If $\overrightarrow a,\:\overrightarrow b\:and\:\overrightarrow c$ are mutually $\perp$ vectors of equal magnitude then the angle $\theta$ made by $\overrightarrow a+\overrightarrow b+\overrightarrow c$with any one of these 3 vectors is ?

$\begin{array}{1 1} cos ^{-1}\large\frac{1}{\sqrt 3} \\ cos ^{-1}\large\frac{1}{ 3} \\ \large\frac{\pi}{2} \\ \large\frac{\pi}{4} \end{array}$

Toolbox:
• Angle between two vectors $\overrightarrow a\:and\:\overrightarrow b$ is given by $cos^{-1}\bigg[\large\frac{\overrightarrow a.\overrightarrow b}{|\overrightarrow a||\overrightarrow b|}\bigg]$
• $|\overrightarrow a+\overrightarrow b+\overrightarrow c|^2=|\overrightarrow a|^2+|\overrightarrow b|^2+|\overrightarrow c|^2+2(\overrightarrow a.\overrightarrow b+\overrightarrow b.\overrightarrow c+\overrightarrow c.\overrightarrow a)$
Given: $|\overrightarrow a|=|\overrightarrow b|=|\overrightarrow c|$ and
$\overrightarrow a,\overrightarrow b \:and\:\overrightarrow c$ are mutually $\perp$.
$\Rightarrow \:\overrightarrow a.\overrightarrow b=\overrightarrow b.\overrightarrow c=\overrightarrow c.\overrightarrow a=0$
Angle between $(\overrightarrow a+\overrightarrow b+\overrightarrow c)\: and\: \overrightarrow a$ is
$cos^{-1}\bigg[\large\frac{(\overrightarrow a+\overrightarrow b+\overrightarrow c).\overrightarrow a}{|\overrightarrow a+\overrightarrow b+\overrightarrow c||\overrightarrow a|}\bigg]$
$=cos^{-1}\bigg[\large\frac{|\overrightarrow a|}{|\overrightarrow a+\overrightarrow b+\overrightarrow c|}\bigg]$
$|\overrightarrow a+\overrightarrow b+\overrightarrow c|^2=3|\overrightarrow a|^2$
Since $\overrightarrow a.\overrightarrow b=\overrightarrow b.\overrightarrow c=\overrightarrow c.\overrightarrow a=0$
and $|\overrightarrow a|=|\overrightarrow b|=|\overrightarrow c|$
$\Rightarrow\:|\overrightarrow a+\overrightarrow b+\overrightarrow c|=\sqrt 3|\overrightarrow a|$
$\therefore$ Required Angle $=cos^{-1}\bigg[\large\frac{|\overrightarrow a|}{\sqrt 3.|\overrightarrow a|}\bigg]$$=cos^{-1}\large\frac{1}{\sqrt 3}$