# The multiplication of a non-zero complex number by i rotates it through

Question:

The multiplication of a non-zero complex number by i rotates it through ____________ in the anti-clockwise direction.

Solution:

Let z = x + iy then iz ix i2y

i.e iz ix – y

i.e iz = – y + ix

$\tan \theta_{1}$ for $z=x+i y, \tan \theta_{1}=\frac{y}{x}$

$\tan \theta_{2}$ for $i z=i x-y, \tan \theta_{2}=\frac{x}{-y}$

Since $\tan \theta_{1} \times \tan \theta_{2}=\frac{y}{x} \times \frac{x}{-y}=-1$

i.e $i z$ is rotated by $90^{\circ}$.