Random unit vectors in n-space

Zathras · **Joined:** Thu, 13 Jul 2006 19:47:54 UTC **Posts:** 308

Let V be the set of vectors in $\Re^n$ s.t. ||v||=1

$\forall v \in V$ under the standard norm. Let $v_1, v_2 \in V$ be random vectors which are outcomes from a uniform distribution on V. What is the probability that the angle between v_1 and v_2 is less than a given angle $\theta$ ?

outermeasure · **Posted:** Fri, 17 Jun 2011 16:55:19 UTC

Zathras wrote:

Let V be the set of vectors in $\Re^n$ s.t. ||v||=1

$\forall v \in V$ under the standard norm. Let $v_1, v_2 \in V$ be random vectors which are outcomes from a uniform distribution on V. What is the probability that the angle between v_1 and v_2 is less than a given angle $\theta$ ?

Think of this as fixing v_1

and choosing a random v_2

, so it is just computation of the volume of ball of radius $\theta$ in the geometry $\mathbb{S}^{n-1}$ , which is
$\displaystyle \int_0^\theta A(r)\,\mathrm{d}r=\frac{\Gamma(\frac{n}{2})}{2\pi^{n/2}} \int_0^\theta \sin^{n-2}r\,\mathrm{d}r.$

S.O.S. Mathematics CyberBoard

Random unit vectors in n-space

Who is online