Argand diagram calculator

Plot it first, then use $\tan\alpha = \left|\frac{b}{a}\right|$ for the angle to the real axis and adjust for the quadrant. For $-2 + 2i$, $\alpha = \frac{\pi}{4}$ and the point is in the second quadrant, so $\arg z = \pi - \frac{\pi}{4} = \frac{3\pi}{4}$.

The principal argument is in $-\pi < \arg z \le \pi$, in radians. A point below the real axis has a negative argument: $\arg(1 - i) = -\frac{\pi}{4}$.

Write $w = re^{i\theta}$. The roots of $z^n = w$ are $r^{1/n}e^{i(\theta + 2k\pi)/n}$ for $k = 0, 1, \dots, n - 1$. For $z^3 = 8i$: $r = 8$ and $\theta = \frac{\pi}{2}$, so the roots are $2e^{i\pi/6}$, $2e^{5i\pi/6}$ and $2e^{-i\pi/2}$, which are $\sqrt{3} + i$, $-\sqrt{3} + i$ and $-2i$.

Every point the same distance from $a$ and $b$: the perpendicular bisector of the line joining them. For $a = 2 - i$ and $b = -1 + 2i$ it is the line $y = x$.