Question

The complex number $z_2$(2 + 3i) is rotated through an angle of ${90}^{\circ }$ anti-clockwise about $z_1$(1 + i). Find the new complex number $z_3$ obtained.

• A. 1 + 2i
• B. -1 -2i
• C. 1-2i
• D. -1 + 2i

Answer 1

The correct option is D

-1 + 2i

Here, we will apply the concept of rotation.

It gives us the relation between the complex numbers $z_1$,$z_2$and $z_3$, which are connected like in the figure.

The following relation can be used to find the any of the complex number, when other two and the angle are given.

We can also find the angle using this relation.

$\frac{z_3-z_1}{z_2-z_1}$ = $\left|\frac{z_3-z_1}{z_2-z_1}\right|$ $e^{i\frac{\pi }{2}}$

$z_3$ - $z_1$ = ($z_2$ - $z_1$ ) $\times$ $e^{i\frac{\pi }{2}}$
$z_3$ - (1+i) = (2+3i -(1+i)) i

= (1+2i)i

$z_3$ - (1+i) = i - 2

$z_3$ = 1+i+(i-2)

= -1+2i