Distance/Point to affine subspace/Example

Let ${\displaystyle {}E}$ be a real affine space over the Euclidean vector space ${\displaystyle {}V}$, let ${\displaystyle {}P\in E}$ be a point, and let ${\displaystyle {}F\subseteq E}$ denote an affine subspace. In case ${\displaystyle {}P\in F}$, the distance of ${\displaystyle {}P}$ to ${\displaystyle {}F}$ equals ${\displaystyle {}0}$. In general, we write

${\displaystyle {}F=Q+U\,}$

with a point ${\displaystyle {}Q\in F}$ and with a linear subspace ${\displaystyle {}U\subseteq V}$. We determine the orthogonal complement ${\displaystyle {}W=U^{\perp }}$ of ${\displaystyle {}U}$ in ${\displaystyle {}V}$. If ${\displaystyle {}u_{1},\ldots ,u_{m}}$ is a basis of ${\displaystyle {}U}$, and ${\displaystyle {}w_{1},\ldots ,w_{k}}$ is a basis of ${\displaystyle {}W}$, then there exists a unique representation

${\displaystyle {}{\overrightarrow {PQ}}=\sum _{i=1}^{m}a_{i}u_{i}+\sum _{j=1}^{k}b_{j}w_{j}\,.}$

In this case,

${\displaystyle {}L=P+\sum _{j=1}^{k}b_{j}w_{j}=Q-\sum _{i=1}^{m}a_{i}u_{i}\,}$

is the dropped perpendicular foot of ${\displaystyle {}P}$ on ${\displaystyle {}F}$, and the distance of ${\displaystyle {}P}$ to ${\displaystyle {}L}$ is

${\displaystyle {}d{\left(P,F\right)}=d{\left(P,L\right)}=\Vert {\sum _{j=1}^{k}b_{j}w_{j}}\Vert \,.}$

If the ${\displaystyle {}w_{j}}$ form an orthonormal basis of ${\displaystyle {}U}$, then this equals ${\displaystyle {}{\sqrt {\sum _{j=1}^{k}b_{j}^{2}}}}$.