Euclidean space/Distance to subspace/Orthogonal projection/Fact

Let ${}V$ be a Euclidean vector space, ${}U\subseteq V$ a linear subspace, and ${}v\in V$.

Then ${}p_{U}(v)$ is the point on ${}U$ that has, among all points of ${}U$, the minimal distance to ${}v$.

In particular, we have

${}d(v,U)=d(v,p_{U}(v))\,.$

Proof, Write another proof