angles in complex plane

Question

There are two points $z_1$ and $z_2$ in the complex plane. What is the angle that the line segment $z_1 z_2$ subtends at the origin?

I want to find this angle in terms of $z_1,z_2$ and $|z_1−z_2|$ and if required, some other angle of the triangle formed but I need to avoid $|z_1|$ and $|z_2|$. Is this possible?

Answer 1

If $z_1 = a_1 e^{i \theta_1}$ and $z_2 = a_2 e^{i \theta_2}$, then $z_1 z_2 = a_1 a_2 e^{i (\theta_1 + \theta_2)}$ and the angle is of course $\theta_1 + \theta_2$. Of course you don't need to use $a_1$ and $a_2$ if you have $\theta_1$ and $\theta_2$.

$$\theta_{12} = \arg (z_1 z_2) = \arg z_1 + \arg z_2$$

I think that is about as basic an answer as is mathematically possible.

Answer 2

By définition: $\arg(z_1z_2)=\arg{z_1}+\arg{z_2}={(\overrightarrow{u},\overrightarrow{M_2M_1})}$ where $\overrightarrow{u}$ represent the unit vector of x-axis and $M_i$ the point of affix $z_i$ in the complex plane.