Complex Numbers (Geometric Representations) What is the geometrical interpretation of this operation:
Multiplication by $\frac{\left(1-i\right)}{\sqrt{2}}$
Attempt:
multiplication by −i = rotate by −π/2

First find the magnitude $r = \left|\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}}i\right| = \sqrt{\frac{1}{2} + \frac{1}{2}} = 1$.
So multiplication by $\frac{\left(1-i\right)}{\sqrt{2}}$ does not change the magnitude.
Next find the argument by $\tan\theta = \frac{-1/\sqrt{2}}{1/\sqrt{2}} = -1 \implies \theta = -\frac{\pi}{4}$.
Thus multiplication by $\frac{\left(1-i\right)}{\sqrt{2}}$ represents a rotation by $\frac{\pi}{4}$ clockwise.
