Students will first learn that if \(L\) is a line in \(\mathbb{R}^2,\) then \(\vec{x} - \text{proj}_L\vec{x}\) is a line perpendicular to \(L.\) After that, students will learn that if \(V\) is a plane in \(\mathbb{R}^3,\) then \(\vec{x} - \text{proj}_V\vec{x}\) is a line perpendicular to \(V.\) Finally, students will be given a definition for the projection of a vector onto a subspace. Here's a lesson on all that.
TODO: Make the figures.