Write $-3i$ in polar coordinates.

Question

can someone please help me with a trivial question?

Write $-3i$ in polar coordinates.

So $z=a+bi=rcis\theta$ with $r=\sqrt{a^{2}+b^{2}}$ and $\theta = arctan\frac{b}{a}$. However, what if $a=0$ such as the case for $-3i$? I am confused!

Answer 1

According to a definition we always have $$-\pi<\theta\le \pi$$here we can write $$\theta=\tan^{-1}\dfrac{-3}{0}=-\dfrac{\pi}{2}\\r=3$$therefore $$-3i=3e^{-i\dfrac{\pi}{2}}$$

Answer 2

Try to draw a picture. I think you will be able to see the angle:

Answer 3

Think of the complex number $z=x+iy$ as a vector from the origin to the point $(x,y)$. Then, it can be characterized by the length $r=|z|$ of the this vector and the angle between the axis $x>0$ and the vector (calculated counterclockwise). Thus, $z=re^{i\theta}$ where $r=\sqrt{x^2+y^2}=3$ and the angle is $-\frac{\pi}{2}$. i.e, $z=3e^{-i\frac{\pi}{2}}$.