Question

u and v are non zero vectors in an inner product space V. Prove that u - proj(v)u is orthogonal to v.

Answer 1

By proj(v)u do you mean projection of v on u or u on v?

Answer 2

u on v.

Answer 3

@cjbrogers do you know the projection formula?

Answer 4

projection of u on v=\[p=\frac{ <u,v>v }{ ||v||^{2} }\]
So, \[<u-p,v>=<u,v>-<p,v>=<u,v>-\frac{ <u,v> }{ ||v||^{2} }<v,v>=<u,v>-<u,v>=0\]

Answer 5

^wow the output came out pretty nicely XD

Answer 6

@nipunmalholtra93, thanks!

Answer 7

np :)