## why are two ranomly choosed vectors in high dimension space likely to be perpandicular to each other? [closed]

I once heard that if you ranomly choosed two vectors in high dimension space, it is very likely that they are perpandicular to each other? Can some give me an intuitive explaination for this? thanks.

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Possibly you misrememeber what you heard; in any case, for most usual interpretations, this seems not true to me. However, this question also seems not appropriate for this site; perhaps reask it on math.stackexchange.com ; yet my advice would be to aks if this is true (as opposed to why). – quid Nov 15 at 14:36
What you are probably thinking of is that the probability that the angle is close to $\pi/2$ is very high as the dimension gets large. If you fix $\delta>0$, and send $n \to \infty$, the probability that two random vectors in $\mathbb{R}^n$ will have an angle between $\pi/2+\delta$ and $\pi/2-\delta$ approaches $1$ as $n \to \infty$. Intuitively, you can already see this on a globe: There is a lot more land between $5^{\circ}$ and $-5^{\circ}$ latitude than between $80^{\circ}$ and $90^{\circ}$. This is a good exercise in multivariable calculus. – David Speyer Nov 15 at 14:54
The OP presumably means "approximately perpendicular" rather than "perpendicular". This might be related to phenomena known as "concentration of measure", but I don't know enough about these to say for sure. – Tobias Fritz Nov 15 at 14:56
If I recall correctly, the probability of an angle between $\alpha$ and $\beta$ is proportional to $\int_{\alpha}^{\beta} \sin^{n-1} \theta d \theta$. In particular, when $\theta$ is not near $\pi/2$, we have $\sin \theta <1$ so the integrand is very small. But check me! – David Speyer Nov 15 at 14:57
Google "Johnson-Lindenstrauss Lemma", Johan, to see that your guess is right. – Bill Johnson Nov 15 at 15:39
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