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Help with Complex Exponentials

  1. Mar 1, 2012 #1
    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. jcsd
  3. Mar 1, 2012 #2

    gneill

    User Avatar

    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##

    Now go ahead and multiply by your e :wink:
     
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