Discover a world of knowledge at Westonci.ca, where experts and enthusiasts come together to answer your questions. Join our platform to connect with experts ready to provide precise answers to your questions in various areas. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.

Which phrase describes how to divide a complex number [tex][tex]$z$[/tex][/tex] by [tex]r(\cos \theta + i \sin \theta)[/tex]?

A. Scale by a factor of [tex]\frac{1}{r}[/tex] and rotate clockwise by [tex]\theta[/tex].
B. Scale by a factor of [tex]r[/tex] and rotate clockwise by [tex]\theta[/tex].
C. Scale by a factor of [tex]\frac{1}{r}[/tex] and rotate counterclockwise by [tex]\theta[/tex].
D. Scale by a factor of [tex]r[/tex] and rotate counterclockwise by [tex]\theta[/tex].


Sagot :

To divide a complex number [tex]\( z \)[/tex] by another complex number, specifically of the form [tex]\( r (\cos \theta + i \sin \theta) \)[/tex], we recall operations on complex numbers in polar form.

1. Complex Number in Polar Form:
- Any complex number [tex]\( z \)[/tex] can be represented in polar form as [tex]\( z = Re^{i\varphi} \)[/tex] where [tex]\( R \)[/tex] is the magnitude (absolute value) and [tex]\( \varphi \)[/tex] is the argument (angle).

2. Dividing Complex Numbers:
- Given two complex numbers [tex]\( z_1 = R_1 e^{i \varphi_1} \)[/tex] and [tex]\( z_2 = R_2 e^{i \varphi_2} \)[/tex], their quotient is given by:
[tex]\[ \frac{z_1}{z_2} = \frac{R_1 e^{i \varphi_1}}{R_2 e^{i \varphi_2}} = \frac{R_1}{R_2} e^{i (\varphi_1 - \varphi_2)} \][/tex]
- This simplifies to scaling the magnitude [tex]\( R_1 \)[/tex] by a factor of [tex]\( \frac{1}{R_2} \)[/tex] and subtracting the angle [tex]\( \varphi_2 \)[/tex] from [tex]\( \varphi_1 \)[/tex].

3. Rephrasing for [tex]\( r (\cos \theta + i \sin \theta) \)[/tex]:
- The complex number [tex]\( r (\cos \theta + i \sin \theta) \)[/tex] can be written in polar form as [tex]\( r e^{i \theta} \)[/tex].
- So, if [tex]\( z = Re^{i\varphi} \)[/tex], then dividing [tex]\( z \)[/tex] by [tex]\( r (\cos \theta + i \sin \theta) \)[/tex] becomes:
[tex]\[ \frac{z}{r (\cos \theta + i \sin \theta)} = \frac{R e^{i \varphi}}{r e^{i \theta}} = \frac{R}{r} e^{i (\varphi - \theta)} \][/tex]

4. Analyzing the Effect:
- Scaling: The magnitude [tex]\( R \)[/tex] is scaled by a factor of [tex]\( \frac{1}{r} \)[/tex].
- Rotation: The angle [tex]\( \varphi \)[/tex] is rotated by [tex]\( -\theta \)[/tex], which is clockwise by [tex]\( \theta \)[/tex] (because subtracting [tex]\(\theta\)[/tex] moves the angle in the clockwise direction).

Thus, the correct description of the effect of dividing a complex number [tex]\( z \)[/tex] by [tex]\( r (\cos \theta + i \sin \theta) \)[/tex] is:

"Scale by a factor of [tex]\(\frac{1}{r}\)[/tex] and rotate clockwise by [tex]\(\theta\)[/tex]."