Altitude of a triangle

An altitude of a triangle is the whole line through one vertex that is orthogonal to the line through the opposite side. For vertex $A$ and opposite vertices $B,C$, its equation in position vectors is $(\mathbf{x}-\mathbf{a})\cdot (\mathbf{c}-\mathbf{b})=0$. The three such lines meet at the orthocenter; they need not meet inside the triangle.