Foot of 3-D Altitude

In this post, we will find a way to project a point onto a plane through the origin. (Planes containing $(0,0,0)^T$ tend to be easier to work with because they form these algebraic structures known as vector spaces. In this case, we are working with the vector space of dimension two over the reals.)

The procedure for projecting the point is very simple if you know how to project a point onto a line.

First, we compute the line $V^\perp$ orthogonal to the plane $V$ at the origin. (We call this line the orthogonal complement in linear algebra.)

Next, we project the point onto $V^\perp$. The projection formula is $$\text{proj}_{V^\perp}v = \frac{v\cdot w}{|w|^2} w$$ where $w\in V^\perp$.

Now note that subtracting the projection from the point gives the projection that we need (try drawing it to see).