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What is geometric interpretation of this equality?

  1. Jan 30, 2015 #1
    1. The problem statement, all variables and given/known data

    Prove that for any a, b ∈ ℂ, |a - b|2 + |a + b|2 = 2(|a|2 + |b|2).

    2. Relevant equations

    |a|2 = aa*

    (a - b)* = (a* - b*)
    (a + b)* = (a* + b*)

    * = complex conjugate

    3. The attempt at a solution

    I've already shown that the relation is true. I'm not quite sure what the geometric interpretation is. My guess is a side of a triangle. -_-
     
    Last edited: Jan 30, 2015
  2. jcsd
  3. Jan 30, 2015 #2
    I believe you can just interpret this as twice the distance from points a and b in the complex plane.
     
  4. Jan 30, 2015 #3

    Mark44

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    Well, you're warm. If a and be represent vectors that are two adjacent sides of a parallelogram, a + b is a vector that represents the long diagonal of the parallelogram. Can you figure out what a - b represents?
     
  5. Jan 30, 2015 #4
    Eh. The short diagonal of parallelogram.
     
  6. Jan 30, 2015 #5

    Mark44

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    Yep.
     
  7. Jan 30, 2015 #6
    So we're taking the norm of those two vectors, squaring them, and adding them.
     
  8. Feb 1, 2015 #7
    Is it some sort of rotation and enlarging of the parallelogram?
     
  9. Feb 1, 2015 #8

    haruspex

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    No.
    Are you seeking a geometrical description for your own interest, or is it part of the question? If the second, is it supposed to match some named theorem?
     
  10. Feb 1, 2015 #9
    It's the last half of the question. I'm not quite sure. I don't see anything in the notes that the professor provides for the class.
     
  11. Feb 1, 2015 #10

    haruspex

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    Ok. So the first step (and perhaps the only step) is to express each of the squared entities as a length. You already have |a-b|, |a+b|. The other two are easier, except that there is a small ambiguity (which you can exploit to make the geometric statement symmetric).
     
  12. Feb 1, 2015 #11
    I've already shown algebraically that the relation is true, using the given equations. That was fairly straightforward. I'm just not (and perhaps I should) seeing the geometric interpretation. o0):headbang:
     
  13. Feb 1, 2015 #12

    haruspex

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    Yes, I understand that, and my post was intended to assist with that. Express each of the squared entities as a length within the diagram. You already have |a-b|, |a+b| as lengths of diagonals of the parallelogram. |a| and |b| are even easier. The only vaguely interesting bit is interpreting the 2 in not quite the obvious way.
     
  14. Feb 1, 2015 #13
    If I'm not mistaken, the aa* and bb* are vectors mapped to the real axis with lengths of a and b squared, respectively.
     
  15. Feb 1, 2015 #14

    haruspex

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    Mapped to the real axis? As in rotated?
    OK, but what are they in terms of the geometry. Don't mention vectors in answering that. They're just squares of lengths of... what?
     
  16. Feb 1, 2015 #15
    You add the angles of the vector and its complex conjugate and the norm is the quantity squared. A square?
     
  17. Feb 1, 2015 #16

    haruspex

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    Did you draw a diagram of the vectors and of the parallelogram that Mark44 referred to?
    (Actually, because this is stated in terms of complex numbers, not vectors, we should be discussing an Argand diagram, but it will come to the same thing.)
     
  18. Feb 1, 2015 #17
    I just redrew it. I'm seeing a few geometric shapes pop out.
     
  19. Feb 1, 2015 #18

    haruspex

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    OK, but you see the parallelogram Mark44 mentioned? Let the vertices in cyclic order be P, Q, R, S. Do you see what distances in that have lengths |a|, |b|, |a-b|, |a+b|?
     
  20. Feb 1, 2015 #19
    Yes, I see the parallelogram and the sides.
     
  21. Feb 1, 2015 #20

    haruspex

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    So express the equation in terms of the lengths PQ etc.
     
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