To determine the product of two complex numbers geometrically, we use polar form to analyze their magnitudes and arguments.

1. Find the magnitudes and arguments of $z = 25 - i$ and $w = \sqrt{3} + 3i$:

   • For $z = 25 - i$:

     • Magnitude $|z|$: $|z| = \sqrt{25^2 + (-1)^2} = \sqrt{625 + 1} = \sqrt{626}$
     • Argument $\arg(z)$: $\arg(z) = \arctan\left(\frac{-1}{25}\right) \approx -\tan^{-1}\left(\frac{1}{25}\right) \approx -2.29^\circ$

   • For $w = \sqrt{3} + 3i$:

     • Magnitude $|w|$: $|w| = \sqrt{(\sqrt{3})^2 + 3^2} = \sqrt{3 + 9} = \sqrt{12} = 2\sqrt{3}$
     • Argument $\arg(w)$: $\arg(w) = \arctan\left(\frac{3}{\sqrt{3}}\right) = \arctan(\sqrt{3}) = 60^\circ$

2. Determine the magnitude and argument of the product $z \cdot w$:

   • The magnitude of $z \cdot w$ is the product of $|z|$ and $|w|$: $|z \cdot w| = |z| \cdot |w| = \sqrt{626} \cdot 2\sqrt{3} = 2\sqrt{1878} = 2\sqrt{3} \cdot |z|$
   • The argument of $z \cdot w$ is the sum of the arguments of $z$ and $w$: $\arg(z \cdot w) = \arg(z) + \arg(w) \approx -2.29^\circ + 60^\circ = 57.71^\circ$

3. Interpret geometrically: To multiply $z$ by $w$, we stretch $z$ by the factor $2\sqrt{3}$ and rotate it by $60^\circ$.

Conclusion:

The correct answer is: "Stretch $z$ by a factor of $2\sqrt{3}$ and rotate $60^\circ$ counterclockwise."

Would you like additional explanations or details on this solution?

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Relative Questions:

1. How are polar forms of complex numbers derived?
2. Why do we add angles when multiplying complex numbers in polar form?
3. What is the geometric interpretation of complex conjugates on the complex plane?
4. How does multiplying by $i$ affect a complex number’s position?
5. What is the effect of multiplying two complex numbers with the same magnitude but different angles?

Tip:

When multiplying complex numbers, remember that their magnitudes multiply, and their arguments add.