Phase of relative coordinate in the complex plane

Question

If we have two points $z_1=x_1+iy_1$ and $z_2=x_2+iy_2$ in the complex plane and define the relative coordinate $z=z_2-z_1$, we have that the length of $z$ is the Euclidian distance between the points:

$r=|z|=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\ .$

But what about the phase of $z$? The best I have been able to come up with is

$\theta=-i\log(z/r)=-i\log\Big(\frac{x_2-x_1+i(y_2-y_1)}{\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}}\Big)\ .$

I kind of expect the answer to be the angle between the points

$\theta_{12}=\cos^{-1}\Big(\frac{\frac{1}{2}(z_1^*z_2+z_1z^*_2)}{|z_1||z_2|}\Big)\ ,$

but I haven't been able to show this.

Answer 1

The phase is $$\theta = \arctan \frac{y_2-y_1}{x_2-x_1}$$ in the respective quadrant.