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Affine geometry

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Homework Statement


[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].

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
 

Answers and Replies

  • #2
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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.
 
  • #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.
 
  • #4
22,097
3,279
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?
That's better.

I feel it's ugly because it largely repeats the construction for existence.
That's exactly why I think it's a pretty argument :smile:
 

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