projection onto a vector

The shadow of $\mathbf u$ cast onto the line through $\mathbf v$ .

The projection of $\mathbf u$ onto $\mathbf v$ , written $\operatorname{proj}_{\mathbf v}(\mathbf u)$ , is $\operatorname{proj}_{\mathbf v}(\mathbf u) = \dfrac{\mathbf u\cdot\mathbf v}{\mathbf v\cdot\mathbf v}\,\mathbf v$ — the piece of $\mathbf u$ that lies exactly along $\mathbf v$ 's direction. Subtracting it off leaves the orthogonal component $\mathbf w = \mathbf u - \operatorname{proj}_{\mathbf v}(\mathbf u)$ , which is always orthogonal to $\mathbf v$ .