Question

The vector represented by the complex number 2 – i is rotated about the origin through an angle $\frac{\mathrm{\pi }}{2}$ in the clockwise direction, the new position of point is

(a) 1 + 2i

(b) –1 –2i

(c) 2 + i

(d) –1 + 2i

Answer 1

Given 2 – i is rotated by $\frac{\mathrm{\pi }}{2}$ angle in the clockwise direction about the origin
$\mathrm{i}.\mathrm{e}\theta =-\frac{\mathrm{\pi }}{2}$
Let Z' denote the new position and Z denote the previous p
$$\begin{gathered} HenceZ\text{'}=Z{e}^{i\theta }=Z{e}^{-\frac{i\pi }{2}} \\ i.eZ\text{'}=\left(2-i\right)\left[\cos \left(\frac{-\pi }{2}\right)+1\sin \left(\frac{-\pi }{2}\right)\right]\left(\because e^{i\theta }=\cos \theta +i\sin \theta \right) \\ i.eZ\text{'}=\left(2-i\right)\left[0+i\left(-1\right)\right] \\ =\left(2-i\right)\left(-i\right) \\ =-2i+i^2 \\ i.eZ\text{'}=-1-2i \end{gathered}$$

Hence, the correct answer is option B.