Let V be an inner product space and $S={v_1,v_2,...v_k}$ be an orthogonal subset of V consisting of nonzero vectors. If $y \in \operatorname{span}(S)$, then $y=\sum_{i=1}^k \frac{\langle y,v_i\rangle}{||v_i||^2} v_i$.
For $y=\sum_{i=1}^k \frac{\langle y,v_i\rangle}{||v_i||^2} v_i$, can someone explain the relation between $\frac{\langle y,v_i\rangle}{||v_i||^2}$ and $\cos \theta =\frac{\langle a,b\rangle}{|a|\cdot |b|}$ ? So does that mean $\frac{\langle y,v_i\rangle}{||v_i||^2}$ is length of orthogonal vector projected onto the subspace A ?