Question-16

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

• \(||x|| = ||y||\)
• \(||x|| > ||y||\)
• \(||x|| < ||y||\)
• There is no relationship between the norms of these two vectors.

Hint

Expand the dot product of \(x - y\) and \(x + y\).

Answer

• \(||x|| = ||y||\)
• \(||x|| > ||y||\)
• \(||x|| < ||y||\)
• There is no relationship between the norms of these two vectors.

Solution

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