Let be two vectors which are not perpendicular. Prove that there exist vector such that is parallel to , is orthogonal to , and .
Proof. Since is parallel to we have for some .
Since is perpendicular to we have
Since we have
Substituting our expression for in terms of and in the last equation we have
Then, substituting this expression for into the equation from perpendicular to we have
which is nonzero since by hypothesis. Thus,
is always a solution