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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?

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closed as off-topic by Carlo Beenakker, Jan-Christoph Schlage-Puchta, Boris Bukh, Mark Wildon, Neil Hoffman Jan 3 at 18:02

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