Complex numbers Locus (1 Viewer)

[oowwoo]

Can someone please give me an explanation on the loci of arg(z-i) = arg(z+2) and arg[(z-i)/z+2)]=pi ?
I know what these loci look like but that's just from memorization, how could I interpret these geometrically?

 

[fan96]

Recall that represents the vector from to .

is the angle that this makes with the positive real axis.

Draw an Argand diagram and mark on it the points and .

Now imagine a variable point moving around the diagram. We want to find the values of for which the vectors and are parallel and facing the same way.

For the second one we can interpret it as finding all for which an angle equivalent to is obtained by subtracting the arguments of the vector from .