  We use the $\mathbf{k} \cdot \mathbf{p}$ theory and the envelope function
approach to evaluate the Rashba spin-orbit coupling induced in a semiconductor
nanowire by a magnetic field at different orientations, taking explicitely into
account the prismatic symmetry of typical nano-crystals. We make the case for
the strongly spin-orbit-coupled InAs semiconductor nanowires and investigate
the anisotropy of the spin-orbit constant with respect to the field direction.
At sufficiently high magnetic fields perpendicular to the nanowire, a 6-fold
anisotropy results from the interplay between the orbital effect of field and
the prismatic symmetry of the nanowire. A back-gate potential, breaking the
native symmetry of the nano-crystal, couples to the magnetic field inducing a
2-fold anisotropy, with the spin-orbit coupling being maximized or minimized
depending on the relative orientation of the two fields. We also investigate
in-wire field configurations, which shows a trivial 2-fold symmetry when the
field is rotated off the axis. However, isotropic spin-orbit coupling is
restored if a sufficiently high gate potential is applied. Our calculations are
shown to agree with recent experimental analysis of the vectorial character of
the spin-orbit coupling for the same nanomaterial, providing a microscopic
interpretation of the latter.
