A man walks a distance of 3 units from the origin towards the North-East (N450E) direction. From there, he walks a distance of 4 units towards the North-West (N450W) direction to reach a point P. Then, the position of P in the Argand plane, is

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A man walks a distance of 3 units from the origin towards the North-East $(N 45^{0} E)$ direction. From there, he walks a distance of 4 units towards the North-West $(N 45^{0} W)$ direction to reach a point $P .$ Then, the position of $P$ in the Argand plane, is

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$(3 - 4 i) e^{i π / 4}$

$3 e^{i π / 4} + 4 i$

$(4 + 3 i) e^{i π / 4}$

$(3 + 4 i) e^{i π / 4}$

answer is D.

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Detailed Solution

Let $O A = 3,$ so that the complex number associated with A is $3 e^{i π / 4} .$ If $z$ is the complex number associated with $P,$ then
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$\frac{z - 3 e^{i π / 4}}{0 - 3 e^{i π / 4}} = \frac{4}{3} e^{- π / 2} = - \frac{4 i}{3} \Rightarrow 3 z - 9 e^{i π / 4} = 12 i e^{i π / 4} \Rightarrow z = (3 + 4 i) e^{i π / 4}$