 Foot of a perpendicular

The foot from $\mathbf{c}$ to the line $\mathbf{a}+t\mathbf{u}$, with $\mathbf{u}\ne 0$, is its orthogonal projection onto that line. Solving $(\mathbf{c}-\mathbf{a}-t\mathbf{u})\cdot \mathbf{u}=0$ gives the displayed formula and proves uniqueness. It belongs to the line segment with endpoints $\mathbf{a}$ and $\mathbf{a}+\mathbf{u}$ exactly when $0\le (\mathbf{c}-\mathbf{a})\cdot \mathbf{u}\le \Vert \mathbf{u}{\Vert }^2$.