Is the orthogonal projection of y onto two orthogonal vectors the value of y?

Question

Is the orthogonal projection of $y$ onto two orthogonal vectors always simply $y$?

I am doing some self-guided study of linear algebra, and something in the course seemed to imply this was the case but I may have been misunderstanding. Is this true?

Answer 1

In $ \mathbb{R}^2 $ if $ u $ and $ v $ are an orthogonal basis, then $ \mathrm{proj}_u{y} + \mathrm{proj}_v{y} = y $. In general, if $ \{ u_n \} $ is an orthogonal basis for the space $ \mathbb{R}^n $ then an analogous statement holds, as I would encourage you to prove.