Theorem: Let $W$ be a finite-dimensional subspace of an inner product space $V$, and let $u \in V$. Then there exists unique vectors $u \in W$ and $z \in W^\perp$ such that $y = u +z $. Furthermore, if $\{ v_1, v_2, ... , v_k\}$ is an orthonormal basis for $W$, then $$ u = \sum_{i=1}^k \langle y, v_i \rangle v_i.$$
Question : $\{ v_1, v_2, ... , v_k\}$ is an orthonormal basis for $W$, how do I interpret those orthonormal basis geometrically? And what does $\langle y, v_i \rangle$ represent geometrically?
Imagine that we are in $\mathbb{R}^3$, and that $W$ is a plane that passes through the origin. Then, $v_1$ and $v_2$ is a pair of orthogonal vectors living in the plane, and, in this case, both has unit length (it doesn't matter for our purpose). Suppose that $y$ is a vector outside the plane. Then, $\langle y,v_1 \rangle v_1$ is the vector that can be obtained projecting $y$ onto $v_1$, and the same for $\langle y,v_2 \rangle v_2$. Also, the linear combination $$u = \langle y,v_1 \rangle v_1 + \langle y,v_2 \rangle v_2$$ represents the "vector" $y$ projected onto the plane. This is why $u$ is called the closest vector to $y$ living in $W$, or the best approximation to $y$ in $W$.