Math Help - Complex number plane multiplication

1. Complex number plane multiplication

We know that multiplying the plane of complex numbers by a constant complex number u multiplies all distances by the absolute value of u.

Assuming that u is not equal to 1, and hence that uz is not equal to z when z is not equal to 0, deduce that multiplication by u is a rotation.

2. Originally Posted by meggnog
We know that multiplying the plane of complex numbers by a constant complex number u multiplies all distances by the absolute value of u.

Assuming that u is not equal to 1, and hence that uz is not equal to z when z is not equal to 0, deduce that multiplication by u is a rotation.
Let $u = \cos{\theta_1} + i\sin{\theta_1}$

and $z = \cos{\theta_2} + i\sin{\theta_2}$.

Therefore $uz = (\cos{\theta_1} + i\sin{\theta_1})(\cos{\theta_2} + i\sin{\theta_2})$

$= \cos{(\theta_1 + \theta_2)} + i\sin{(\theta_1 + \theta_2)}$.

Since the angle has changed, that means you have a rotation.