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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