# Relating the arguments of two complex numbers

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

• What are you confused about? $arg(ki) = \frac{\pi}{2}$. Arg is the angle the complex number makes relative to the positive real axis. – ImHereSometimes Jan 15 '18 at 17:18
• Oh yes ! Didn’t think about it that way ! Thank you :) – Aditi Jan 15 '18 at 17:20

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 $\pi/2$.
• Thank you , but could you please tell if the above statement implies that the angle between ${Z_2}$ and ${Z_1}$ is $\frac{\pi}{2}$ ? – Aditi Jan 15 '18 at 17:33
Since $\frac{Z_1}{Z_2} = ki$ for $k\in \mathbb{R^+}$ is purely imaginary its argument is $\frac{\pi}{2}$.
$\frac {Z_1}{Z_2}=re^{i\theta}=ki \implies e^{i\theta}=i\implies \theta =\frac \pi 2$, since $e^{i\theta}=\cos\theta +i\sin\theta$...