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(D) Let the origin be $O(0, 0)$. The man walks $3$ units in the direction of $N 45^{\circ} E$,which corresponds to an angle of $\pi / 4$ with the posi

Vedclass · Accepted Answer

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

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(D) Let the origin be $O(0, 0)$. The man walks $3$ units in the direction of $N 45^{\circ} E$,which corresponds to an angle of $\pi / 4$ with the positive $x$-axis. The position of point $A$ is $z_A = 3 e^{i \pi / 4}$.From $A$,he walks $4$ units in the direction of $N 45^{\circ} W$. The direction $N 45^{\circ} W$ makes an angle of $45^{\circ} + 90^{\circ} = 135^{\circ}$ or $3\pi / 4$ with the positive $x$-axis.The displacement vector from $A$ to $P$ is $4 e^{i 3\pi / 4}$.Thus,the position of $P$ is $z_P = z_A + 4 e^{i 3\pi / 4} = 3 e^{i \pi / 4} + 4 e^{i 3\pi / 4}$.Since $e^{i 3\pi / 4} = e^{i \pi / 4} \cdot e^{i \pi / 2} = i e^{i \pi / 4}$,we have:$z_P = 3 e^{i \pi / 4} + 4 i e^{i \pi / 4} = (3 + 4 i) e^{i \pi / 4}$.

$A$ 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

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