Proving Uniqueness of Affine Plane Containing S & Weak-Parallel to T

[A_B]

Homework Statement

S and T are two affine lines in \mathbb{A}^3 that are not parallel and S\cap T=\emptyset.

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

The Attempt at a Solution

Existence is easy, if S=p+V and T=q+W then R=p+(V+W) satisfies the conditions.

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

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

Thanks
Alex

 

[micromass]

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.

 

[A_B]

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.

 

[micromass]

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