  Higgs multiplet in the vector-spinor representations of SO(10), i.e., the
$144+\bar{144}$ multiplet can break the SO(10) gauge symmetry spontaneously in
one step down to the Standard Model gauge group symmetry $SU(3)_C\times
SU(2)_L\times U(1)_Y$ and a recent analysis has used such vector-spinors for
building a new class of SO(10) grand unification models (hep-ph/0506312) . Here
we discuss the techniques for the computation of several classes of
vector-spinor couplings using the Basic Theorem on the SO(2N) vertex expansion
developed by the authors. The computations include the cubic couplings of the
vector-spinors with SO(10) tensors, quartic self-couplings of the
vector-spinors, and couplings of the vector-spinors with spinor representations
of SO(10). The last set include couplings of vector-spinors with the 16-plets
of quarks and lepton and with the 16 and $\bar{16}$ of Higgs. These couplings
provide a crucial tool for further development of the SO(10) grand unification
using vector-spinor representations. These include study of quark-lepton
masses, analysis of dimension five operators including baryon and lepton number
violating operators, and study of neutrino masses and mixings. Illustrative
examples are given for their computation using a sample of vector-spinor
couplings.
