We compare the magnetic field orientation for the young giant molecular cloud Vela C inferred from 500-$mu$m polarization maps made with the BLASTPol balloon-borne polarimeter to the orientation of structures in the integrated line emission maps from Mopra observations. Averaging over the entire cloud we find that elongated structures in integrated line-intensity, or zeroth-moment maps, for low density tracers such as $^{12}$CO and $^{13}$CO $J$ $rightarrow$ 1 - 0 are statistically more likely to align parallel to the magnetic field, while intermediate or high density tracers show (on average) a tendency for alignment perpendicular to the magnetic field. This observation agrees with previous studies of the change in relative orientation with column density in Vela C, and supports a model where the magnetic field is strong enough to have influenced the formation of dense gas structures within Vela C. The transition from parallel to no preferred/perpendicular orientation appears to happen between the densities traced by $^{13}$CO and by C$^{18}$O $J$ $rightarrow$ 1 - 0. Using RADEX radiative transfer models to estimate the characteristic number density traced by each molecular line we find that the transition occurs at a molecular hydrogen number density of approximately $10^3$ cm$^{-3}$. We also see that the Centre-Ridge (the highest column density and most active star-forming region within Vela C) appears to have a transition at a lower number density, suggesting that this may depend on the evolutionary state of the cloud.
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