Relating the arguments of two complex numbers

Question

If $Z_1$ and $Z_2$ are two complex numbers such that $\frac{Z_1}{Z_2} = ki$, then why do we say that $arg\frac{Z_1}{Z_2} = \frac{\pi}{2} $ ?

Answer 1

I assume $k$ is real here. write $Z_i = r_1 e^{i\phi_2}$ and then $Z_1/Z_2 = r_1/r_1 \times e^{i(\phi_1-\phi_2)}$ You want this to be purely imaginary hence $\phi_1-\phi_2 = \pi/2 (+ 2\pi n)$. The angle between the two is indeed $\pi/2$.

Answer 2

Since $\frac{Z_1}{Z_2} = ki$ for $k\in \mathbb{R^+}$ is purely imaginary its argument is $\frac{\pi}{2}$.

Answer 3

$\frac {Z_1}{Z_2}=re^{i\theta}=ki \implies e^{i\theta}=i\implies \theta =\frac \pi 2 $, since $e^{i\theta}=\cos\theta +i\sin\theta $...