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Equivalence relation (geometry)

  1. Jan 20, 2014 #1
    1. The problem statement, all variables and given/known data

    Let ##\mathbb{R}^2 = \{Q = (a,b) | a,b\in \mathbb{R}\}##. Prove that if ##q_1 = (a_1,b_1)## and ##q_2=(a_2,b_2)## are equivalent, meaning ##a_1^2+b_1^2 = a_2^2 +b_2^2##, then this gives an equivalence relation on ##\mathbb{R}^2##. What is ##[(1,0)], [(0,1)],[(2,2)],[(0,0)]?## What does an equivalence class look like?


    2. The attempt at a solution

    I know how to do the first part with the equivalence relation but I am not sure how to do the second part of the question?
     
  2. jcsd
  3. Jan 20, 2014 #2

    Dick

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    I'm not sure what you are asking about. I assume [(0,1)] means the equivalence class of (0,1). What does that look like?
     
  4. Jan 20, 2014 #3
    Yes, it is the equivalence class but I don't understand how it looks like. Is it just a unit circle?
     
  5. Jan 20, 2014 #4

    Dick

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    Well, yes. If (x,y) is related to (0,1) then x^2+y^2=0^2+1^2=1. That's the equation of the unit circle.
     
  6. Jan 20, 2014 #5
    Gotcha and that goes for (1,0) which will be a circle centered at 0 with radius 1 and how about (2,2)?
     
  7. Jan 20, 2014 #6

    Dick

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    It's pretty similar to the other one, isn't it? You tell me what kind of circle it is.
     
  8. Jan 21, 2014 #7
    I understand now. Thanks for the help!
     
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