Let $A$ be an [affine space](/page/Affine%20Space) modeled on a [vector space](/page/Vector%20Space) $V$ over a field $k$. Let $W \subset V$ be a linear subspace, let $p \in A$, and define $B := p+W = \{p+w : w \in W\}$. Then $\operatorname{dir}(B)=W$. In particular, $\operatorname{dir}(B)$ is a linear subspace of $V$, and for every $r \in B$, one has $B=r+\operatorname{dir}(B)$.