  We examine the transverse and longitudinal components of the drag force upon
a straight vortex line due to the scattering of liquid $^4$He excitations. For
this purpose, we consider a recently proposed Hamiltonian that describes the
dissipative motion of a vortex, giving an explicit expression for the
vortex-quasiparticle interaction. The involved dissipative coefficients are
obtained in terms of the reservoir correlation function. Most of our explicit
calculations are concerned to the range of temperatures below 0.4 K, at which
the reservoir is composed by phonon quasiparticle excitations. We also discuss
some important implications in the determination of possible scattering
processes leading to dissipation, according to the values of vortex mass found
in the literature.
