Question

A man walks a distance of 3 units, from the origin towards the north - east $\left(N{45}^{\circ }E\right)$ direction. From there, he walks a distance of 4 units towards the north - west $\left(N{45}^{\circ }E\right)$ direction to reach a point P. Then the position of P in the Argand plane is:

• A. ${3e}^{i\pi /4}+4i$
• B. ${(3-4i)e}^{i\pi /4}$
• C. ${(4+3i)e}^{i\pi /4}$
• D. ${(3+4i)e}^{i\pi /4}$

Answer 1

The correct option is D

${(3+4i)e}^{i\pi /4}$

$\overline{OP}=\overline{OA}+\overline{AP}$
$⟹$ $\overline{OP}=\overline{OA}+\overline{OB}$
$⟹\overline{OP}={3e}^{i\pi /4}+{4e}^{i3\pi /4}={3e}^{i\pi /4}+{4e}^{i(\pi /2+\pi /4)}={3e}^{i\pi /4}+{4e}^{i\pi /2}.e^{i\pi /4}$
$={3e}^{i\pi /4}+i{4e}^{i\pi /4}=(3+4i)e^{i\pi /4}$