Position of P in Argand Plane

Question

A man walks a distance of 3 units from the origin towards the north-east direction. From there, he walks a distance of 4 units towards the north-west 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) $(3+4i)e^{i\pi /4}$.

Let O be origin. Let A be 3 units from O in the north-east direction. So, A is $3e^{i\pi /4}$. From here, distance to P is 4 units in north-west. So, I guess P should on Y-axis, with co-ordinates being $(0,5)$. $\triangle OAP$ being right-angled at A. So, my answer is $5i$. But it doesn't match with the given options.

Answer 1

First shift the plane by $3e^{iπ/4},$ then again by $4e^{i3π/4}.$ Thus we have moved the origin in all by $$3e^{iπ/4}+4e^{i3π/4}=e^{iπ/4}(3+4e^{iπ/2})=e^{iπ/4}(3+4i).$$

Answer 2

Consider the journey as $\vec{OA}+\vec{AP}=\vec{OP}$.

Then $OA$ is given by $z=3e^{i\pi/4}$. Going northwest from $A$ means we have rotated counterclockwise by $90^{\circ}$, which can be achieved by multiplying a complex number by $e^{i\pi/2}=i$. So for $AP$, we will have $\frac{4}{3}iz$. Now to get $OP$ we just have to add $$OP=z+\frac{4}{3}iz=3e^{i\pi/4}+4i(e^{i\pi/4})=(3+4i)e^{i\pi/4}.$$