Question

If u and v are orthogonal unit vectors in an inner product space, then ||u-v|| = _______. Justify your answer with appropriate computations.

Answer 1

\(||u-v||= |u|^2+|v|^2\)
Proof:
||u-v||= <u-v,u-v>
=<u-v,u> - <u-v,v>
=<u,u> - <v,u> - <u,v> + <v,v>
u and v are orthogonal, therefore <u,v>=<v,u>=0
so that \( ||u-v||= <u,u>+<v,v>=|u|^2+|v|^2\)

Answer 2

Thank you