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

$\text{First: } 3 \text{ units from } O \text{ along } N\,45^\circ E$

$\text{then: } 4 \text{ units along } N\,45^\circ W \text{ to reach } P$

$\text{To find: position of } P \text{ in the Argand plane}$

$1.\ \text{write the two displacements as complex numbers}$ $2.\ \text{add them}$ $3.\ \text{match with the options}$ $N\,45^\circ E \Rightarrow 45^\circ \text{ from the positive real axis}$ $z_1 = 3e^{i\pi/4}$ $z_1 = 3(\cos45^\circ + i\sin45^\circ)$ $= \dfrac{3}{\sqrt2} + i\,\dfrac{3}{\sqrt2}$ $N\,45^\circ W \Rightarrow 135^\circ \text{ from the positive real axis}$ $z_2 = 4e^{i3\pi/4}$ $e^{i3\pi/4}=e^{i\pi/2}\cdot e^{i\pi/4}$ $= i\,e^{i\pi/4}$ $z_2 = 4i\,e^{i\pi/4}$ $P = z_1 + z_2 = 3e^{i\pi/4} + 4i\,e^{i\pi/4}$ $P = (3+4i)\,e^{i\pi/4}$ $P = (3+4i)\,e^{i\pi/4}$ $\text{Answer: option (4)}$