Question

which statement describes how to geometrically divide a complex number, z, by a second complex number, w?

• scale z by the modulus of w, then rotate clockwise by the argument of w.
• scale z by the modulus of w, then rotate counterclockwise by the argument of w.
• scale z by the reciprocal of the modulus of w, then rotate clockwise by the argument of w.
• scale z by the reciprocal of the modulus of w, then rotate counterclockwise by the argument of w.

Explanation:

When working with complex numbers in polar form, if $z = r_1(\cos\theta_1 + i\sin\theta_1)$ and $w = r_2(\cos\theta_2 + i\sin\theta_2)$, then $\frac{z}{w} = \frac{r_1}{r_2}(\cos(\theta_1 - \theta_2) + i\sin(\theta_1 - \theta_2))$. This means we scale $z$ by the reciprocal of $w$'s modulus ($\frac{1}{r_2}$) and rotate $z$ clockwise by $w$'s argument (since subtracting the argument is equivalent to a clockwise rotation).

Answer:

Scale z by the reciprocal of the modulus of w, then rotate clockwise by the argument of w.