Is Multiplying a Complex Number by ejθ a Positive Rotation of Its Vector?

[JJHK]

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

j² = -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_o² = a² + b²
L_o = √(a²+b²)

Length of new z, L₁:

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

Therefore L_o = 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!

 

[gneill]

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

cosθ + jsinθ = e^jθ

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 e^j$\phi$

Now go ahead and multiply by your e^jθ