Question

let $z = 3(cos(15^{circ}) + i sin(15^{circ}))$ and $w = 5(cos(90^{circ}) + i sin(90^{circ}))$. what best describes the geometric construction of the quotient $\frac{z}{w}$ on the complex plane?

• $z$ is scaled by a factor of $\frac{1}{5}$ and rotated 90 degrees clockwise.
• $z$ is scaled by a factor of 1 and rotated 90 degrees clockwise.
• $z$ is scaled by a factor of 1 and rotated 90 degrees counterclockwise.
• $z$ is scaled by a factor of $\frac{1}{5}$ and rotated 90 degrees counterclockwise.

Explanation:

Step1: Recall complex quotient rule

For complex numbers $z=r_1(\cos\theta_1+i\sin\theta_1)$ and $w=r_2(\cos\theta_2+i\sin\theta_2)$, $\frac{z}{w}=\frac{r_1}{r_2}(\cos(\theta_1-\theta_2)+i\sin(\theta_1-\theta_2))$

Step2: Calculate scaling factor

$\frac{r_1}{r_2}=\frac{3}{5}$? No, wait: scaling $z$ to get $\frac{z}{w}$: $\frac{z}{w}=z\times\frac{1}{w}$, $\frac{1}{w}=\frac{1}{5}(\cos(-90^\circ)+i\sin(-90^\circ))$, so scaling factor is $\frac{1}{5}$

Step3: Calculate rotation angle

$\theta_1-\theta_2=15^\circ-90^\circ=-75^\circ$, which is equivalent to rotating $z$ by $-90^\circ$ (clockwise 90°) plus adjusting, but the rotation component from dividing by $w$ is rotating $z$ clockwise by 90° (since subtracting 90° is clockwise rotation on complex plane)

Answer:

z is scaled by a factor of $\frac{1}{5}$ and rotated 90 degrees clockwise.