1. ## Complex numbers Locus

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?

2. ## Re: Complex numbers Locus

Recall that $z-w$ represents the vector from $w$ to $z$.

$\arg (z-w)$ is the angle that this makes with the positive real axis.

Draw an Argand diagram and mark on it the points $-2$ and $i$.

Now imagine a variable point $z$ moving around the diagram. We want to find the values of $z$ for which the vectors $z-i$ and $z-(-2)$ are parallel and facing the same way.

For the second one we can interpret it as finding all $z$ for which an angle equivalent to $\pi$ is obtained by subtracting the arguments of the vector $z-(-2)$ from $z-i$.

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