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Complex Vector Question

  1. Nov 2, 2012 #1
    1. The problem statement, all variables and given/known data

    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


    2. Relevant equations
    tan(θ1)=b/a
    tan(θ2)=d/c
    |Z1|=Z1
    |Z2|=Z2

    3. 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, θZ is the sum of these two. I must prove:

    θZ=(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(θZ)=imaginary/real

    I get tan(θZ)=(ad+bc)/(ac-bd).

    I'm not quite sure what I'm doing wrong, but I'd really appreciate a hand! Thanks!
     
  2. jcsd
  3. Nov 2, 2012 #2

    TSny

    User Avatar
    Homework Helper
    Gold Member

    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(θZ).
     
  4. Nov 2, 2012 #3
    Have you been introduced to Euler's formula yet or are you required to solve it in cartesian form?
     
  5. Nov 2, 2012 #4
    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.
     
  6. Nov 2, 2012 #5

    Many thanks,

    Using the relation tan(θ12)=(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.

    :smile:
     
  7. Nov 2, 2012 #6
    A lot easier to solve in polar form :)
     
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