Rotation by angle alpha about the origin: z -> z * e^(i*alpha)

Multiplying by e^(i*alpha) rotates the point by alpha counterclockwise.

Rotation by angle alpha about point z0: z -> z0 + (z - z0) * e^(i*alpha)

Spiral similarity: Multiplying by re^(ialpha) simultaneously scales by r and rotates by alpha. This is the geometric meaning of complex multiplication.

Application to collinearity and perpendicularity:

• z1, z2, z3 are collinear iff $\frac{z3-z1}{(z2-z1)}$ is real
• z1z2 perpendicular to z3z4 iff $\frac{z2-z1}{(z4-z3)}$ is purely imaginary

JEE application: "If z1, z2, z3 form an equilateral triangle, then $z1^2$ + $z2^2$ + $z3^2$ = z1z2 + z2z3 + z3*z1." This comes from the rotation: z3-z1 = (z2-z1)*e^$\frac{i*pi}{3}$.