Physics - Vectors Question with Solution | TestHub

From the above diagram, the equations are, 
$\overrightarrow{\mathrm{OA}}=R\left[\cos {30}^{\circ }\hat{\mathrm{i}}+\sin {30}^{\circ }\hat{\mathrm{j}}\right]$

$\overrightarrow{\mathrm{OB}}=R\left[\cos {60}^{\circ }\hat{\mathrm{i}}-\sin 60°\hat{\mathrm{j}}\right]$

$\overrightarrow{\mathrm{OC}}=R\left[-\cos 45°\hat{\mathrm{i}}+\sin 45°\hat{\mathrm{j}}\right]$

$\Rightarrow \overrightarrow{\mathrm{OA}}+\overrightarrow{\mathrm{OB}}-\overrightarrow{\mathrm{OC}}=R\left[\frac{1}{2}+\frac{\sqrt{3}}{2}+\frac{1}{\sqrt{2}}\right]\hat{\mathrm{i}}+R\left[\frac{1}{2}-\frac{\sqrt{3}}{2}-\frac{1}{\sqrt{2}}\right]\hat{\mathrm{j}}$, angle with the $x-$axis.

$\Rightarrow \tan \alpha =\frac{\frac{1}{2}-\frac{\sqrt{3}}{2}-\frac{1}{\sqrt{2}}}{\frac{1}{2}+\frac{\sqrt{3}}{2}+\frac{1}{\sqrt{2}}}=\frac{1-\sqrt{3}-\sqrt{2}}{1+\sqrt{3}+\sqrt{2}}$

$\Rightarrow \alpha ={\tan }^{-1}\left[\frac{1-\sqrt{3}-\sqrt{2}}{1+\sqrt{3}+\sqrt{2}}\right]$.