Rotation formula Complex numbers

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  • #1
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Homework Statement


[tex] If arg(\frac{z-ω}{z-ω^2}) = 0, \ then\ prove \ that\ Re(z) = -1/2 [/tex]

Homework Equations


ω and ω^2 are non-real cube roots of unity.

The Attempt at a Solution


arg(z-ω) = arg(z-ω^2)
So, z-ω = k(z-w^2)
Beyond that, I'm not sure how to proceed. Using the rotation formula may also be required, but I'm not sure how to use it here.
Furthermore, I can sort of intuitively visualise the answer, but I'm not sure how to prove it analytically.
 
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Answers and Replies

  • #2
SammyS
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Homework Statement


[tex] \text{If}\ \arg \left( \frac{z-ω}{z-ω^2} \right) = 0, \ \text{then prove that }\ Re(z) = -1/2 [/tex]

Homework Equations


ω and ω^2 are non-real cube roots of unity.

The Attempt at a Solution


arg(z-ω) = arg(z-ω^2)
So, z-ω = k(z-w^2)
Beyond that, I'm not sure how to proceed. Using the rotation formula may also be required, but I'm not sure how to use it here.
Furthermore, I can sort of intuitively visualise the answer, but I'm not sure how to prove it analytically.
If ω is one of the non-real cube roots of unity, then what is ω2, and how is it related to ω ?
 
  • #3
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1 + ω + ω^2 = 0
[tex] ω = e^\frac{i2\pi}{3}[/tex]
How does that help?
 
  • #4
SammyS
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1 + ω + ω^2 = 0
[tex] ω = e^\frac{i2\pi}{3}[/tex]
How does that help?
I was thinking in terms of the complex conjugate.
 
  • #5
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ω is the conjugate of ω^2.
 
  • #6
Dick
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You have that (z-ω)=k(z-ω^2) where k is real and positive. Take the real part of both sides.
 
  • #7
haruspex
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Your target is Re(z). How else can you write that?
 
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