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Affine geometry

  1. Nov 1, 2011 #1

    A_B

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    1. The problem statement, all variables and given/known data
    [itex]S[/itex] and [itex]T[/itex] are two affine lines in [itex]\mathbb{A}^3[/itex] that are not parallel and [itex]S\cap T=\emptyset[/itex].

    Show there is a unique affine plane [itex]R[/itex] that contains [itex]S[/itex] and is weak parallel with [itex]T[/itex].

    3. The attempt at a solution
    Existence is easy, if [itex]S=p+V[/itex] and [itex]T=q+W[/itex] then [itex]R=p+(V+W)[/itex] satisfies the conditions.

    To prove uniqueness I assume planes [itex]R[/itex] and [itex]Q[/itex] both satisfy all conditions. They both contain [itex]S[/itex] so they can be written as [itex]p+(V+vectorspace)[/itex]. That vectorspace must be [itex]W[/itex] since the planes must be weak parallel with [itex]T[/itex]so both [itex]R[/itex] and [itex]Q[/itex] are equal to [itex]p+(V+W)[/itex].

    Is this good?
    If it is, it still seems very ugly to me, is there a better way to do it?

    Thanks
    Alex
     
  2. jcsd
  3. Nov 1, 2011 #2

    micromass

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    What is ugly about the solution?? It seems nice...

    There is a little detail missing though. For uniqueness, you must use somewhere that S and T are not parallel.
     
  4. Nov 1, 2011 #3

    A_B

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    Thanks micromass!

    Ok,The direction of R and Q must contain W, since V does not contain W, and is not a subspace of W (S and T are not parrallel and have dimensions 1) , "vectorspace" must be W
    good now?

    I feel it's ugly because it largely repeats the construction for existence.

    thanks again.
     
  5. Nov 1, 2011 #4

    micromass

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    That's better.

    That's exactly why I think it's a pretty argument :smile:
     
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