Euclidean space/Affine subspace/Perpendicular/Fact

Let ${\displaystyle {}V}$ be a Euclidean vector space, and let ${\displaystyle {}E_{1}=P_{1}+U_{1}}$ and ${\displaystyle {}E_{2}=P_{2}+U_{2}}$ denote nonempty affine subspaces with the linear subspaces ${\displaystyle {}U_{1},U_{2}\subseteq V}$. Let

${\displaystyle {}P_{1}-P_{2}=u_{1}+u_{2}+u\,}$

with ${\displaystyle {}u_{1}\in U_{1}}$, ${\displaystyle {}u_{2}\in U_{2}}$, and ${\displaystyle {}u\in {\left(U_{1}+U_{2}\right)}^{\perp }}$.

Then the

distance ${\displaystyle {}d(E_{1},E_{2})}$ equals ${\displaystyle {}\Vert {u}\Vert }$; it is obtained in the points ${\displaystyle {}P_{1}-u_{1}\in E_{1}}$ and

${\displaystyle {}P_{2}+u_{2}\in E_{2}}$.

In particular, the connecting vector of the points, where the minimal distance is obtained, is perpendicular to ${\displaystyle {}E_{1}}$ and to ${\displaystyle {}E_{2}}$.