I know the proof for two random vectors$\def\a{\mathbf{a}} \def\b{\mathbf{b}} \def\c{\mathbf{c}}$ (say $\a$ and $\b$) in high dimension becomes orthogonal, i.e. $\langle \a,\b\rangle = 0$. I am keen to know what happens when $\a$ and $\b$ are correlated. Can they be quasi-orthogonal? In that case, how would be the inner product be defined?
Clarification: I am following the proof shown here. The statement clearly says that both the vectors are randomly drawn (as a result I assume there is no correlation involved). But in my case, I have $\|\a−\c\|_2=\|\b\|_2.$ Now, I want to know whether in high dimension, $\a$ and $\b$ can be treated as orthonormal or not. Can this be proven that they are orthogonal (if they are)?
Best Answer
Since $a=c+b$ and $c\cdot b=0$, $a\cdot b=b^2\ge0$ and $a\cdot c=c^2\ge0$.