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]."
