# Affine geometry

1. Nov 1, 2011

### A_B

1. The problem statement, all variables and given/known data
$S$ 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 $T$.

3. 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 $T$so 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

2. Nov 1, 2011

### micromass

Staff Emeritus
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. Nov 1, 2011

### 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.

4. Nov 1, 2011

### micromass

Staff Emeritus
That's better.

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