  We construct a bipartite generalization of Alon and Szegedy's nearly
orthogonal vectors, thereby obtaining strong bounds for several extremal
problems involving the Lov\'asz theta function, vector chromatic number,
minimum semidefinite rank, nonnegative rank, and extension complexity of
polytopes. In particular, we derive a couple of general lower bounds for the
vector chromatic number which may be of independent interest.
