If you really are "completely lost" then the following won't likely help you much and, perhaps, it is now time to think about private lessons and/or going through the material thoroughly.

U is a plane in the real euclidean space (in this case, it is a plane through the origin and thus it is a subspace) , and its orthogonal complement is the subspace of all vectors that are perpendicular to it.

About (orthogonal) projections onto subspace of vectors you may want to read about orthonormal basis and linear projections: this is way too lengthy to describe it here.

Tonio