let $P,P'$ two affine subspace of $R^{3}$
have we equality between this two statement
$$\exists\ u_{0}\in R^{3}\ \mbox{such that } t_{u_0}(P)=P'$$
$$\exists B,A\in PP' \mbox{such that } u_{0}=\vec{BA}(=A-B) \ \mbox{ and } t_{u_0}(B)=A $$
any help would be appreciated
Make a drawing!
Assume that $P,P'$ are two planes. $t_{u_0}(P)=P'$ iff {$P,P'$ are parallel and there are $B\in P,A\in P'$ s.t. $u_0=\vec{BA}$}. Note that $u_0=\vec{BA}$ iff $t_{u_0}(B)=A$. Finally the first assertion implies the second one ; yet the converse is false.