# Homework Help: Help with Complex Exponentials

1. Mar 1, 2012

### JJHK

1. The problem statement, all variables and given/known data

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 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.

2. Relevant equations

z = a + jb

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

cosθ + jsinθ = e

j2 = -1

3. The attempt at a solution

so z(e)
= (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, Lo:

Lo2 = a2 + b2
Lo = √(a2+b2)

Length of new z, L1:

L12 = (acosθ - bsinθ)2 + (asinθ + bcosθ)2
L12 = a2 + b2
L1 = √(a2+b2)

Therefore Lo = L1

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!

2. Mar 1, 2012

### Staff: Mentor

Have you considered that your complex number can also be described in exponential form using the same relationship:

cosθ + jsinθ = e

if R = |a2 + b2| is the magnitude of the complex number, and $\phi$ = atan(b/a) is the argument of the complex number (equivalent vector's direction angle), then the vector r for the complex number can also be represented by:

r = R(cos$\phi$ + jsin$\phi$) = R ej$\phi$