Vector $A$ having magnitude $\sqrt{2}$ units makes an angle 45${}^{\circ }$ with horizontal and vector $B$ having magnitude $2$ units makes an angle 30${}^{\circ }$ with horizontal as shown in the figure. What is $\overrightarrow{A}+\overrightarrow{B}$?

The correct option is A (1 + $\sqrt{3}$) $\hat{i}$ + 2 $\hat{j}$
Taking components the vector $A$ can be written as
$\overrightarrow{A}$ = ($\sqrt{2}$ cos (45${}^{\circ }$) $\hat{i}$ + $\sqrt{2}$ sin (45${}^{\circ }$) $\hat{j}$ )
= $\hat{i}$ + $\hat{j}$
Taking components the vector $B$ can be written as
$\overrightarrow{B}$ = (2 cos (30 ${}^{\circ }$) $\hat{i}$ + 2 sin (30${}^{\circ }$) $\hat{j}$ )
= $\sqrt{3}$ $\hat{i}$ + $\hat{j}$

Hence,
$\overrightarrow{A}$ + $\overrightarrow{B}$
= ($\hat{i}$ + $\hat{j}$) + ($\sqrt{3}$ $\hat{i}$ + $\hat{j}$)
= (1 + $\sqrt{3}$) $\hat{i}$ + 2 $\hat{j}$