Question

which statement describes how to geometrically determine the product of $z = 25 - i$ and $w = sqrt{3} + 3i$ on the complex plane?\
\
\bigcirc stretch $z$ by a factor of $sqrt{3}$ and rotate $30^circ$ counterclockwise.\
\bigcirc stretch $z$ by a factor of $sqrt{3}$ and rotate $60^circ$ counterclockwise.\
\bigcirc stretch $z$ by a factor of $2sqrt{3}$ and rotate $30^circ$ counterclockwise.\
\bigcirc stretch $z$ by a factor of $2sqrt{3}$ and rotate $60^circ$ counterclockwise.

Explanation:

Step1: Find modulus of $w$

For complex number $w = a+bi$, modulus is $\sqrt{a^2+b^2}$.
$$|w| = \sqrt{(\sqrt{3})^2 + 3^2} = \sqrt{3+9} = \sqrt{12} = 2\sqrt{3}$$

Step2: Find argument of $w$

Argument $\theta$ satisfies $\tan\theta = \frac{b}{a}$.
$$\tan\theta = \frac{3}{\sqrt{3}} = \sqrt{3} \implies \theta = 60^\circ$$

Step3: Match to geometric operation

Geometrically, multiplying $z$ by $w$ stretches $z$ by $|w|$ and rotates by $\arg(w)$.

Answer:

Stretch z by a factor of $2\sqrt{3}$ and rotate $60^\circ$ counterclockwise.