let $z = 3 + 2i$ and $w = 4 + i$. what best describes the geometric construction of the product $z$ times…

let $z = 3 + 2i$ and $w = 4 + i$. what best describes the geometric construction of the product $z$ times $w$ on the complex plane?\n$\\circ$ $z$ is stretched by a factor of $\\sqrt{17}$ and rotated $14^\\circ$ counterclockwise\n$\\circ$ $z$ is stretched by a factor of $\\sqrt{17}$ and rotated $76^\\circ$ counterclockwise\n$\\circ$ $z$ is stretched by a factor of 5 and rotated $14^\\circ$ counterclockwise\n$\\circ$ $z$ is stretched by a factor of 5 and rotated $76^\\circ$ counterclockwise

let $z = 3 + 2i$ and $w = 4 + i$. what best describes the geometric construction of the product $z$ times $w$ on the complex plane?\n$\\circ$ $z$ is stretched by a factor of $\\sqrt{17}$ and rotated $14^\\circ$ counterclockwise\n$\\circ$ $z$ is stretched by a factor of $\\sqrt{17}$ and rotated $76^\\circ$ counterclockwise\n$\\circ$ $z$ is stretched by a factor of 5 and rotated $14^\\circ$ counterclockwise\n$\\circ$ $z$ is stretched by a factor of 5 and rotated $76^\\circ$ counterclockwise

Answer

Explanation:

Step1: Find modulus of $w$

The modulus of a complex number $a+bi$ is $\sqrt{a^2+b^2}$. For $w=4+i$: $\vert w\vert = \sqrt{4^2+1^2} = \sqrt{16+1} = \sqrt{17}$

Step2: Find argument of $w$

The argument $\theta$ of $a+bi$ is $\tan^{-1}\left(\frac{b}{a}\right)$. For $w=4+i$: $\arg(w) = \tan^{-1}\left(\frac{1}{4}\right) \approx 14^\circ$

Step3: Interpret geometric product rule

When multiplying complex numbers $z$ and $w$, $z$ is stretched by $\vert w\vert$ and rotated by $\arg(w)$ counterclockwise.

Answer:

z is stretched by a factor of $\sqrt{17}$ and rotated $14^\circ$ counterclockwise