Question

A man walks a distance of $3\text{units,}$ from the origin towards the north - east $\left(N{45}^{\circ }E\right)$ direction. From there, he walks a distance of $4\text{units}$ 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. ${(3+4i)e}^{(\frac{i\pi }{4})}$
• B. ${(3-4i)e}^{(\frac{i\pi }{4})}$
• C. ${(4+3i)e}^{(\frac{i\pi }{4})}$
• D. ${3e}^{(\frac{i\pi }{4})}+4i$

Answer 1

The correct option is A

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

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