To answer this question, we will express each number in its polar form.
1) The polar form of z is:
[tex]z=r\cdot e^{i\theta},[/tex]
where r is the distance from the origin, and θ is its angle in radians measured counterclockwise from the x-axis. From the figure, we see that z is 180° from the origin, that angle is θ = π. So the polar form of z is:
[tex]z=r\cdot e^{i\pi}.[/tex]
2) The cartesian components of the second number are:
[tex]x+iy=-2+2i\Rightarrow x=-2,y=2.[/tex]
To find its polar form, we represent it in the plane:
We see that this number lies in the second quadrant. Angles in the second quadrant are given by the following formula:
[tex]\theta=\tan ^{-1}(\frac{y}{x})+\pi=\tan ^{-1}(\frac{2}{-2})+\pi=\tan ^{-1}(-1)+\pi=-\frac{\pi}{4}+\pi=\frac{3}{4}\pi.[/tex]
The r coordinate is given by:
[tex]r=\sqrt[]{x^2+y^2}=\sqrt[]{(-2)^2+2^2}=\sqrt[]{2\cdot4}=2\cdot\sqrt[]{2.}[/tex]
So the polar form of the second angle is:
[tex]-2+2i=\sqrt[]{2^2+2^2}\cdot e^{i\cdot\tan ^{-1}(2/-2)}=2\cdot\sqrt[]{2}\cdot e^{i\cdot\frac{3}{4}\pi}.[/tex]
3) Now, we multiply the numbers in their polar form:
[tex](z)\cdot(-2+2i)=(r\cdot e^{i\pi})\cdot(2\cdot\sqrt[]{2}\cdot e^{i\cdot\frac{3}{4}\pi})=(2\cdot\sqrt[]{2r})\cdot r\cdot e^{i\pi+i\frac{3}{4}\pi}=(2\cdot\sqrt[]{2})\cdot r\cdot e^{i\cdot\frac{7}{4}\pi}.[/tex]
Converting to degrees the angle of the resulting number, we get:
[tex]\frac{7}{4}\pi=\frac{7}{4}\pi\cdot\frac{360^{\circ}}{2\pi}=315^{\circ}\text{.}[/tex]
The distance from the origin of the resulting number is:
[tex]2\cdot\sqrt[]{2}\cdot r\cong2.82\cdot r\text{.}[/tex]
So the resulting angle has:
• a magnitude (distance from the origin) approximately 2.82 the magnitude of z,
,
• an angle θ = 315°.
From the points of the figure, the only one that meets these conditions is point D, which is at an angle θ = 315° and at a distance that is 3r, being r the distance of z from the origin.
Answer: D
