Problem-14

Question

\(x\) and \(y\) are vectors in \(\mathbb{R}^{n}\). If \(x-y\) is orthogonal to \(x+y\), comment on the relationship between \(||x||\) and \(||y||\).

Solution

If \(x-y\) and \(x+y\) are orthogonal, then we have \(\langle x-y,x+y\rangle =0\). From this we get \(\langle x,x\rangle =\langle y,y\rangle \Longrightarrow ||x||=||y||\).