Thread: argument of a complex number

If |z1 + z2| = |z1 - z2|, prove that the difference between the arguments (of z1 and z2) is pi/2.

I'd be grateful for any help or pointers in trying to prove this please. Thanks.

write them down as exponential funcitons: z1=r1*exp(i*phi1)
calculate the |z1+z2| by multiplying by the same with -i instead of i
r1 and r2 all fall out
it stays exp(i*(phi1-phi2)) = - exp(-i*(phi1-phi2))
divide by exp(-i*(phi1-phi2))
now it's exp(i*2*(phi1-phi2)) = -1
so phi1-phi2 must be pi/2 because exp(i*2*pi/2)=exp(i*pi)=-1