How do I find the half-line equation for the locus $\operatorname{arg}(z - a) = \frac{-3\pi}{4}$?

Question

So there are a locus composed of points $m$ that is $z$ and it's a half-line:

$\operatorname{arg}(z - a) = \frac{-3\pi}{4}$

$a = - 1 + i$

The problem is that I have no idea how to find out the equation(Half-line equation I mean)

Answer 1

This is the half-line from $a= (a_x, a_y)$ in the direction $\theta= -\frac{3\pi}{4}$. So the Cartesian equation is $y- a_y= m(x- a_x)$ where m is the slope of the line: $m= tan(3\pi/4)= $$\frac{sin(-3\pi/4)}{cos(-3\pi/4)}=$$ -\frac{sin(3\pi/4)}{cos(3\pi/4)}$.

$sin(3\pi/4)= \frac{\sqrt{2}}{2}$ and $cos(3\pi/4)= -\frac{\sqrt{2}}{2}$ so this is -1.