Question

Suppose we multiply a complex number z by-2 + 2i.ImZRe.ABoODWhich point represents the product of z and -2 + 2i?Choco 1 ano

Answer 1

To answer this question, we will express each number in its polar form.

1) The polar form of z is:

`z=r\cdot e^(i\theta),`

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:

`z=r\cdot e^(i\pi).`

2) The cartesian components of the second number are:

`x+iy=-2+2i\Rightarrow x=-2,y=2.`

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:

`\theta=\tan ^(-1)((y)/(x))+\pi=\tan ^(-1)((2)/(-2))+\pi=\tan ^(-1)(-1)+\pi=-(\pi)/(4)+\pi=(3)/(4)\pi.`

The r coordinate is given by:

`r=\sqrt[]{x^2+y^2}=\sqrt[]{(-2)^2+2^2}=\sqrt[]{2\cdot4}=2\cdot\sqrt[]{2.}`

So the polar form of the second angle is:

`-2+2i=\sqrt[]{2^2+2^2}\cdot e^{i\cdot\tan ^(-1)(2/-2)}=2\cdot\sqrt[]{2}\cdot e^{i\cdot(3)/(4)\pi}.`

3) Now, we multiply the numbers in their polar form:

`(z)\cdot(-2+2i)=(r\cdot e^(i\pi))\cdot(2\cdot\sqrt[]{2}\cdot e^{i\cdot(3)/(4)\pi})=(2\cdot\sqrt[]{2r})\cdot r\cdot e^{i\pi+i(3)/(4)\pi}=(2\cdot\sqrt[]{2})\cdot r\cdot e^{i\cdot(7)/(4)\pi}.`

Converting to degrees the angle of the resulting number, we get:

`(7)/(4)\pi=(7)/(4)\pi\cdot(360^(\circ))/(2\pi)=315^(\circ)\text{.}`

The distance from the origin of the resulting number is:

`2\cdot\sqrt[]{2}\cdot r\cong2.82\cdot r\text{.}`

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