Rays in the space

Question

I have a nice problem from a mathematical circle: Let n be a positive integer. Determine the smallest n with the property in the space having

Answer 1

I guess that with "space" you mean euclidean three-space ${\mathbb R}^3$.

Assume that we are given $n$ unit vectors $x_i$ attached at $0\in{\mathbb R}^3$, with no acute angle between any two of them. The tips of these $x_i$ lie on the unit sphere $S^2$, and any two of these tips have a spherical distance $\geq{\pi\over2}$. It follows that each of these tips can be considered as center of a spherical cap on $S^2$ of spherical radius $\rho:={\pi\over4}$, and no two of these caps overlap. The ${\mathbb R}^3$-height of such a cap is given by $h=1-\cos\rho=1-{1\over\sqrt{2}}$; its area therefore computes to $2\pi\bigl(1-{\sqrt{2}\over2}\bigr)$. It follows that $$n\cdot 2\pi\left(1-{\sqrt{2}\over2}\right)\leq4\pi\ ,$$ which implies $n\leq 2(2+\sqrt{2})<6.83$, or $n\leq6$.

On the other hand one can find six vectors without an acute angle between any two of them: Let $(e_1,e_2,e_3)$ be an orthonormal basis of ${\mathbb R}^3$, and consider the six vectors $\pm e_i$.

It follows that the minimal $n$ guaranteeing a pair of vectors including an acute angle is $7$.