Question

what does multiplying a complex number by a real number do to its polar representation?

Answer 1

It multiplies both the real and imaginary parts by the same real factor. This increases the length r in the polar vector (r, theta) representation, but not its angle, theta.

Any complex number can be written
x + iy or r e^(i*theta) where r cos theta is the real part, x , and r sin theta is the imaginary part, iy. x and y are real