Vector angle lab

Why vectors gather near 90°

Start with a sphere you can rotate. Then raise the dimension and watch the angle distribution squeeze toward a right angle.

Representation dimensions 3D Visible feature vectors 96 Near-90° band 89°–91°

3D intuition

Rotate the vector sphere

Literal 3D

Loading the 3D sphere…

query q inside angle band outside band

Distribution

Where do the angles land?

Expected in band —

This sample —

Spread around 90° —

What to notice

See the compact math

For unit vectors, q · f = cos θ. For uniformly random directions in d dimensions, the relative angle density is proportional to sin^d−2(θ). Raising d therefore concentrates the mass around 90°.