  We consider a large condensate in a rotating anisotropic harmonic trap. Using
the method of matched asymptotic expansions, we derive the velocity of an
element of vortex line as a function of the local gradient of the trap
potential, the line curvature and the angular velocity of the trap rotation.
This velocity yields small-amplitude normal modes of the vortex for 2D and 3D
condensates. For an axisymmetric trap, the motion of the vortex line is a
superposition of plane-polarized standing-wave modes. In a 2D condensate, the
planar normal modes are degenerate, and their superposition can result in
helical traveling waves, which differs from a 3D condensate. Including the
effects of trap rotation allows us to find the angular velocity that makes the
vortex locally stable. For a cigar-shape condensate, the vortex curvature makes
a significant contribution to the frequency of the lowest unstable normal mode;
furthermore, additional modes with negative frequencies appear. As a result, it
is considerably more difficult to stabilize a central vortex in a cigar-shape
condensate than in a disc-shape one. Normal modes with imaginary frequencies
can occur for a nonaxisymmetric condensate (in both 2D and 3D). In connection
with recent JILA experiments, we consider the motion of a straight vortex line
in a slightly nonspherical condensate. The vortex line changes its orientation
in space at the rate proportional to the degree of trap anisotropy and can
exhibit periodic recurrences.
