Question

which description of the transformation of z on the complex plane gives the product of $z = \sqrt{2}\left(\cos\left(\frac{\pi}{4}\
ight) + i\sin\left(\frac{\pi}{4}\
ight)\
ight)$ and $w = \sqrt{8}\left(\cos\left(\frac{\pi}{4}\
ight) + i\sin\left(\frac{\pi}{4}\
ight)\
ight)$? \
\
\bigcirc scale z by a factor of 4, then rotate counterclockwise $\frac{\pi}{2}$ radians \
\
\bigcirc scale z by a factor of $\sqrt{8}$, then rotate counterclockwise $\frac{\pi}{2}$ radians \
\
\bigcirc scale z by a factor of $\sqrt{8}$, and then rotate counterclockwise $\frac{\pi}{4}$ radians \
\
\bigcirc scale z by a factor of 4, then rotate counterclockwise $\frac{\pi}{4}$ radians

Explanation:

Step1: Recall complex multiplication rule

For complex numbers in polar form $z=r_1(\cos\theta_1 + i\sin\theta_1)$ and $w=r_2(\cos\theta_2 + i\sin\theta_2)$, their product is $zw=r_1r_2(\cos(\theta_1+\theta_2)+i\sin(\theta_1+\theta_2))$.

Step2: Calculate scaling factor

Find $r_1r_2$: $r_1=\sqrt{2}$, $r_2=\sqrt{8}$, so $r_1r_2=\sqrt{2}\times\sqrt{8}=\sqrt{16}=4$.

Step3: Calculate rotation angle

Find $\theta_1+\theta_2$: $\theta_1=\frac{\pi}{4}$, $\theta_2=\frac{\pi}{4}$, so $\theta_1+\theta_2=\frac{\pi}{4}+\frac{\pi}{4}=\frac{\pi}{2}$.

Step4: Match to transformation

Scaling factor is 4, rotation is $\frac{\pi}{2}$ counterclockwise.

Answer:

scale z by a factor of 4, then rotate counterclockwise $\frac{\pi}{2}$ radians