Complex Multiplication/Division

Given:
$z_1=a_1+jb_1=\left|z_1\right|e^j\theta_1$
$z_2=a_2+jb_2=\left|z_2\right|e^j\theta_2$

Q1: What is $z_1z_2$ in Cartesian form?
A1: Using the FOIL method and collecting the real and imaginary parts:
$z_1z_2=(a_1a_2-b_1b_2)+j(a_1b_2+a_2b_1)$

Q2: What is $z_1z_2$ in polar form?
A2: $z_1z_2=\left|z_1\right|\left|z_2\right|e^{j(\theta_1+\theta_2)}$

Q3: What is $\frac{z_1}{z_2}$ in Cartesian form?
A3: Multiplying both the numerator and denominator by $z_2^*$:
$\frac{z_1}{z_2}=\frac{(a_1a_2+b_1b_2)+j(a_2b_1-a_1b_2)}{a_2^2+b2_2}$

Q4: What is $\frac{z_1}{z_2}$ in polar form?
A4: $\frac{z_1}{z_2}=\frac{\left|z_1\right|}{\left|z_2\right|}e^{j(\theta_1-\theta_2)}$