TITLE: Why are two "random" vectors in $\mathbb R^n$ approximately orthogonal for large $n$?
QUESTION [20 upvotes]: I saw that two random independent vectors are approximately orthogonal in high dimensional space.
How can I prove this?
And is there an intuitive explanation?
Thank you.

REPLY [4 votes]: One way to come at this is to try to stretch your intuition even farther, toward the Johnson-Lindenstrauss lemma, which says that while we can only fit $n$ orthogonal vectors into $\mathbb{R}^n$, we can easily (randomly) fit exponentially many almost-orthogonal vectors.  The precise statement of the lemma includes the result you mention as a special case, so it's not really fair to say it provides the intuition.  Rather I mention it because if you can fit it into your toolbox for thinking about high dimensions it will make other things more intuitive.