  Specific features of two-dimensional nematodynamics give rise to shortfalls
of the tensor representation of the nematic order parameter commonly used in
computations, especially in theory of active matter. The alternative
representation in terms of the vector order parameter follows with small
adjustments the classical director-based theory, but is applicable to 2D
problems where both nematic alignment and deviation from the isotropic state
are variable. Stability analysis of nematic alignment and flow is used as a
testing ground. A director-based analysis demonstrates a shortfall of the
standard theory, which does not ensure relaxation to equilibrium in a passive
system. It also demonstrates the inadequacy of the director-based description,
which misses a stabilizing effect of perturbations of the modulus ensuring
stability of a passive system on scales far exceeding the healing length.
