Perpendicular vectors out of a set
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.