  Magnetic fields play an important role in shaping and regulating star
formation in molecular clouds. Here, we present one of the first studies
examining the relative orientations between magnetic ($B$) fields and the dust
emission, gas column density, and velocity centroid gradients on the 0.02 pc
(core) scales, using the BISTRO and VLA+GBT observations of the NGC 1333
star-forming clump. We quantified these relative orientations using the Project
Rayleigh Statistic (PRS) and found preferential global parallel alignment
between the $B$ field and dust emission gradients, consistent with large-scale
studies with Planck. No preferential global alignments, however, are found
between the $B$ field and velocity gradients. Local PRS calculated for
subregions defined by either dust emission or velocity coherence further
revealed that the $B$ field does not preferentially align with dust emission
gradients in most emission-defined subregions, except in the warmest ones. The
velocity-coherent structures, on the other hand, also showed no preferred $B$
field alignments with velocity gradients, except for one potentially
bubble-compressed region. Interestingly, the velocity gradient magnitude in NGC
1333 ubiquitously features prominent ripple-like structures that are indicative
of magnetohydrodynamic (MHD) waves. Finally, we found $B$ field alignments with
the emission gradients to correlate with dust temperature and anticorrelate
with column density, velocity dispersion, and velocity gradient magnitude. The
latter two anticorrelations suggest that alignments between gas structures and
$B$ fields can be perturbed by physical processes that elevate velocity
dispersion and velocity gradients, such as infall, accretions, and MHD waves.
