Question:
The vector represented by the complex number $2-i$ is rotated about the origin through an angle $\frac{\pi}{2}$ in the clockwise direction, the new position of point is
(a) 1 + 2i
(b) –1 –2i
(c) 2 + i
(d) –1 + 2i
Solution:
Given $2-\mathrm{i}$ is rotated by $\frac{\pi}{2}$ angle in the clockwise direction about the origin
i. e $\theta=-\frac{\pi}{2}$
Let $Z^{\prime}$ denote the new position and $Z$ denote the previous $p$
Hence $Z^{\prime}=Z e^{i \theta}=Z e^{-\frac{i \pi}{2}}$
i.e Z' $=(2-i)\left[\cos \left(\frac{-\pi}{2}\right)+1 \sin \left(\frac{-\pi}{2}\right)\right] \quad\left(\because e^{i \theta}=\cos \theta+i \sin \theta\right)$
i.e $Z^{\prime}=(2-i)[0+i(-1)]$
$=(2-i)(-i)$
$=-2 i+i^{2}$
i.e $Z^{\prime}=-1-2 i$
Hence, the correct answer is option B.