Perpendicular vectors out of a set

I stumbled on this problem and I wanna know if there is a better solution.
There are $n$ 3d vectors with $x$, $y$, and $z$ components and I wanna find all pairs of perpendicular vectors in this set. Is there any better way than to verify all possible pairs of 3 vectors, some kind of optimization (not the obvious ones)?

The problem can be solved in time $\tilde{O}(n^{4/3})$, using several algorithms:

• Agarwal, Partitioning arrangements of lines II: Applications.

• Chazelle, Cutting hyperplanes for divide-and-conquer.

• Matoušek, Range searching with efficient hierarchical cuttings.

There is an essentially matching lower bound due to Erickson, New lower bounds for Hopcroft's problem. See also Williams and Yu, Finding orthogonal vectors in discrete structures.

StackExchange Computer Science Q#113601, answer score: 5