Question

How to prove a complex number lies on a straight line?

Answer 1

If in a complex plane z1 is represented by point A,z2 by B and z3 by C, then z3−z2 represents the vector →BC, and z2−z1 represents vector →AB. Thus, these points A,B,C are collinear iff →AB and →BCare parallel as they already have one point B in common ⟹→BC=c→AB⟹z3−z2=c(z2−z1).

Here, c is taken real because when we multiply by a complex no. , it's real part gives the scaling and imaginary part rotates it. Here, we need the vectors to be parallel ,so we omit the rotation and c has to be real only.

hope it works