To solve this problem, we need to use the properties of multiplying complex numbers in polar form. When two complex numbers are in polar form, their product is obtained by multiplying their magnitudes and adding their angles.

Given:

• $z = 8 \left( \cos \frac{\pi}{3} + i \sin \frac{\pi}{3} \right)$
• $w = 3 \left( \cos \frac{\pi}{6} + i \sin \frac{\pi}{6} \right)$

Let's find the magnitude and argument of $z \cdot w$.

Step 1: Magnitude of $zw$

The magnitude of $z$ is $|z| = 8$. The magnitude of $w$ is $|w| = 3$.

The magnitude of the product $zw$ is: $|zw| = |z| \cdot |w| = 8 \cdot 3 = 24$

Step 2: Argument of $zw$

The argument of $z$ is $\arg(z) = \frac{\pi}{3}$. The argument of $w$ is $\arg(w) = \frac{\pi}{6}$.

The argument of the product $zw$ is: $\arg(zw) = \arg(z) + \arg(w) = \frac{\pi}{3} + \frac{\pi}{6} = \frac{\pi}{2}$

Conclusion

The product $zw$ has a magnitude of 24 and an argument of $\frac{\pi}{2}$. Thus, on the complex plane, this transformation is represented by stretching $z$ by a factor of 3 and rotating by $\frac{\pi}{2}$ radians counterclockwise.

Answer

The correct answer is:

• Stretch $z$ by a factor of 3 and rotate $\frac{\pi}{2}$ radians counterclockwise.

Would you like additional details or have any questions?

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Here are five additional related questions:

1. How would the construction change if we rotated $w$ by $\frac{\pi}{4}$ radians instead?
2. How can we express $zw$ in rectangular form?
3. What would happen to the construction if $|w|$ was doubled?
4. How is the rotation angle affected if the angle of $z$ changes?
5. Can we interpret the rotation direction (counterclockwise) in terms of the unit circle?

Tip: When multiplying complex numbers in polar form, always remember to multiply the magnitudes and add the angles for an efficient solution.