Distance between a vector and a subspace

Question

Let $V$ be a vector space of finite dimension, and let $0 \neq u \in V$. Denote $U=(span \text{{u}})^\bot$.

Prove that for every $v \in V$

$$dist(v,U)=\frac {\lvert \left\langle v,u \right\rangle\rvert}{\lVert u\rVert}$$

Answer 1

The distance form $v$ to $U$ is the distance from $v$ to the only $w\in U$ such that $v-w$ is orthogonal to $U$, that is, such that $v-w=\lambda u$, for some $\lambda\in\mathbb R$. Since $v-\frac{\langle v,u\rangle}{\|u\|^2}u\in U$, you just have to take $\lambda=\frac{\langle v,u\rangle}{\|u\|^2}$. So, $w=v-\lambda u=v-\frac{\langle v,u\rangle}{\|u\|^2}u$ and the distance from $v$ to $w$ is equal to $\frac{\bigl|\langle v,u\rangle\bigr|}{\|u\|}$.