Proving the Angle Relationship in Complex Vector Multiplication

[nucleawasta]

Homework Statement

Question from Vibrations and Waves by A.P. French Chapter 1

Consider a vector z defined by Z=Z₁Z₂, where Z₁=a+jb, Z₂=c+jd.

a)Show that the length of the of z is the product of the lengths of Z₁ and Z₂.

b)Show that the angle between z and the x-axis is the sum of of the angles made by Z₁ and Z₂

Homework Equations

tan(θ₁)=b/a
tan(θ₂)=d/c
|Z₁|=Z₁
|Z₂|=Z₂

The Attempt at a Solution

So the first part I didn't have any trouble with, it was fairly straight forward showing that the length of Z₁*Z₂ was equal to the length of Z. But when I moved to part B I ran into a problem. Here's what I tried.

I Knew θ₁=b/a and θ₂=d/c by a first order taylor expansion of the tangents of these angles and since I am told the angle of Z, θ_Z is the sum of these two. I must prove:

θ_Z=(cb+da)/ca

However when I write out the form of Z=Z₁Z₂ I get:

Z=ac-bd +j(ad+bc). Now since I know the tan(θ_Z)=imaginary/real

I get tan(θ_Z)=(ad+bc)/(ac-bd).

I'm not quite sure what I'm doing wrong, but I'd really appreciate a hand! Thanks!

 

[TSny]

You don't want to make any small angle approximation since you want to prove it for arbitrary angles.

tan(θ_Z)=(ad+bc)/(ac-bd).

Try finding a trig identity involving the tangent function that you can relate to your expression for tan(θ_Z).

 

[NemoReally]

Have you been introduced to Euler's formula yet or are you required to solve it in cartesian form?

 

[nucleawasta]

I mean I'm actually a senior physics major :P(slightly embarrassing I couldn't solve this) I've seen Euler's identity and it is introduced in the chapter, so I suppose that could be a viable way to solve the problem.

 

[nucleawasta]

You don't want to make any small angle approximation since you want to prove it for arbitrary angles.

Try finding a trig identity involving the tangent function that you can relate to your expression for tan(θ_Z).

Many thanks,

Using the relation tan(θ₁+θ₂)=(tan(θ₁)+tan(θ₂))/1-tan(θ₁)*tan(θ₂)

I was able to use trigonometry(SOHCAHTOA as i learned it way back when) to plug in for the tan(θ₁) and tan(θ₂) which ultimately leads to the solution I was trying to prove from my first post.

 

[aralbrec]

A lot easier to solve in polar form :)