If $X_1$ and $X_2$ are two independent isotropic random vectors in $\mathbb{R}^n$, then $\mathbb{E}\|X_i\|_{2}^{2}=n$, $\mathbb{E}\langle X_1,X_2\rangle^{2}=n$.
How can I show from the above result that in high dimensional spaces, independent and isotropic random vectors are almost orthogonal?
