Two random unit vectors on a sphere, Geometric Probability

Problem 28.1

Let $\mathbf{u}$ and $\mathbf{v}$ be two independent uniformly-distributed unit vectors on the sphere $S^2 \subset \mathbb{R}^3$ . What is the expected value of $|\mathbf{u} \cdot \mathbf{v}|$ ?

Solution. By rotational symmetry, we may condition on $\mathbf{v}$ and assume, without loss of generality, that $\mathbf{v} = (0, 0, 1)$ . Then $\mathbf{u} \cdot \mathbf{v} = u_3$ , the third coordinate of $\mathbf{u}$ .

For $\mathbf{u}$ uniformly distributed on $S^2$ , its third coordinate $u_3$ has a well-known distribution: $u_3$ is uniform on $[-1, 1]$ . This is the content of Archimedes’s hat-box theorem — the sphere’s “height” coordinate has uniform density with respect to Lebesgue measure on the interval. (Equivalently: the vertical strip $\{ \mathbf{u} \in S^2 : h \leq u_3 \leq h + dh \}$ has the same area for every $h \in [-1, 1]$ , namely $2\pi \, dh$ .)

With $u_3 \sim \operatorname{Uniform}[-1, 1]$ , the expected absolute value is $\mathbb{E}[|u_3|] = \int_{-1}^{1} |t| \cdot \tfrac12 \, dt = \int_0^1 t \, dt = \tfrac12.$ Therefore $\boxed{\mathbb{E}[|\mathbf{u} \cdot \mathbf{v}|] = \tfrac12.} \qedhere$

The result is a clean one-line consequence of Archimedes’s hat-box theorem, with no heavy calculation. In contrast, the two-dimensional analogue — $\mathbf{u}, \mathbf{v}$ iid uniform on the unit circle $S^1$ — gives $\mathbb{E}[|\mathbf{u} \cdot \mathbf{v}|] \;=\; \mathbb{E}[|\cos \Theta|] \;=\; \frac{2}{\pi},$ where $\Theta =$ angle between them is uniform on $[0, \pi]$ . The appearance of $\pi$ in 2D versus the rational $\tfrac12$ in 3D is a consequence of the different angular Jacobian on the circle ( $d\theta / 2\pi$ ) versus on the sphere (hat-box: uniform on height).

import numpy as np
# Uniform on S^2 via normalised Gaussian
u = np.random.randn(3, 10**7); u /= np.linalg.norm(u, axis=0)
v = np.random.randn(3, 10**7); v /= np.linalg.norm(v, axis=0)
dots = np.abs((u*v).sum(axis=0))
print(f"sim: {dots.mean():.5f}   exact: 0.5")
# sim: 0.49989   exact: 0.5