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Complex analysis

  1. Jan 27, 2009 #1
    1. The problem statement, all variables and given/known data
    Show that the vector z1 is parallel to z2 if and only if Im(z1z2*)=0

    note: z2* is the complement of z2


    2. Relevant equations



    3. The attempt at a solution
    I would probably convert z to polar form.
    so, z1=r1(cos Ѳ1+isin Ѳ1)
    z2=r2(cos Ѳ2+isin Ѳ2)
    so, z2*=r2(cos Ѳ2-isin Ѳ2)

    Then, I would plug it into Im(z1z2*)=0

    so, Im(r1(cos Ѳ1+isin Ѳ1)r2(cos Ѳ2-isin Ѳ2))

    which is: r1r2sin Ѳ2sin Ѳ2

    But I'm not sure where to go from here....

    PLEASE HELP!!

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



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Jan 27, 2009 #2

    Dick

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    Your final answer for the imaginary part isn't correct. If you do it right, it might just resemble the trig addition formula for sin(theta1-theta2).
     
  4. Jan 27, 2009 #3
    Ok, so I recalculated my final value for the imaginary part and got...

    0=r1r2(sin Ѳ1cos Ѳ2-cos Ѳ1sin Ѳ2)

    So then I got:
    sin Ѳ1cos Ѳ2=cos Ѳ1sin Ѳ2

    So from here, do I just try to show that Ѳ12 to show that the vectors are parallel?
    If so, how would I go about doing that?
     
  5. Jan 27, 2009 #4

    Dick

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    Doesn't that look like a trig formula for the sine of the difference of two angles?
     
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