- #1

JJHK

- 24

- 1

## Homework Statement

Hello all, I'm currently reading through "Vibrations and Waves" by A.P French. I'm not very used to these type of books, so I could use a little bit of help right now:

**Show that the multiplication of any complex number z by e**^{jθ}is describable, in geometrical terms, as a positive rotation through the angle θ of the vector by which z is represented, without any alteration of its length.## Homework Equations

z = a + jb

j = an instruction to perform a counterclockwise rotation of 90 degrees upon whatever it precedes.

cosθ + jsinθ = e

^{jθ}

j

^{2}= -1

## The Attempt at a Solution

so z(e

^{jθ})

= (a + jb)(cosθ + jsinθ)

= acosθ + (asinθ + bcosθ)j - bsinθ

= (acosθ - bsinθ) + (asinθ + bcosθ)j

Now I'm supposed to show that the above equation is

*a positive rotation through the angle θ of the vector by which z is represented, without any alteration of its length.*

First, I'm going to prove that the length was not altered:

*Original length of z, L*

_{o}:L

_{o}

^{2}= a

^{2}+ b

^{2}

L

_{o}= √(a

^{2}+b

^{2})

*Length of new z, L*

_{1}:L

_{1}

^{2}= (acosθ - bsinθ)

^{2}+ (asinθ + bcosθ)

^{2}

L

_{1}

^{2}= a

^{2}+ b

^{2}

L

_{1}= √(a

^{2}+b

^{2})

Therefore L

_{o}= L

_{1}

Now, How do I prove the rotation? I'm not too strong with trig, maybe that's the problem? Can somebody help me out? Thank you!