Vector projection (${\mathrm{proj}}_{\mathbf{c}}\mathbf{a}=(\mathbf{a}\cdot \mathbf{c})\mathbf{c}/\Vert \mathbf{c}{\Vert }^2$)

The vector projection multiplies the scalar projection by the chosen unit vector, giving the orthogonal projection onto the linear span of a nonzero real Euclidean vector $\mathbf{c}$:For $\mathbf{a}=(3,4)$ and $\mathbf{c}=(2,0)$, this gives $\mathbf{p}=(6/4)(2,0)=(3,0)$ and $\mathbf{r}=(0,4)$; the dot product $\mathbf{r}\cdot \mathbf{c}=0$ checks the orthogonal projection residual. Replacing $\mathbf{c}$ by $(-2,0)$ gives the same $\mathbf{p}=(-6/4)(-2,0)=(3,0)$: reversing direction changes the scalar projection, but preserves the vector projection onto the line.

For a short proof of the nearest-point property, any vector $\mathbf{z}$ on this line satisfies $\Vert \mathbf{a}-\mathbf{z}{\Vert }^2=\Vert \mathbf{r}{\Vert }^2+\Vert \mathbf{p}-\mathbf{z}{\Vert }^2$ by the Pythagorean identity. The second term is uniquely minimized at $\mathbf{z}=\mathbf{p}$. This connects a directional measurement to the orthogonal projection used in approximation.^[ref]