Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 15E / a / Solution

Orthogonal projection gives the unique componentsChooseThen $a,b\ge 0$, $a^2+b^2=1$, and the required decomposition follows. If one projection vanishes, its coefficient is zero and the corresponding normalized vector may be chosen arbitrarily in that subspace; the nonzero projected components remain unique.