  We present results for Vela C obtained during the 2012 flight of the
Balloon-borne Large Aperture Submillimeter Telescope for Polarimetry
(BLASTPol). We mapped polarized intensity across almost the entire extent of
this giant molecular cloud, in bands centered at 250, 350, and 500 {\mu}m. In
this initial paper, we show our 500 {\mu}m data smoothed to a resolution of 2.5
arcminutes (approximately 0.5 pc). We show that the mean level of the
fractional polarization p and most of its spatial variations can be accounted
for using an empirical three-parameter power-law fit, p = p_0 N^(-0.4)
S^(-0.6), where N is the hydrogen column density and S is the
polarization-angle dispersion on 0.5 pc scales. The decrease of p with
increasing S is expected because changes in the magnetic field direction within
the cloud volume sampled by each measurement will lead to cancellation of
polarization signals. The decrease of p with increasing N might be caused by
the same effect, if magnetic field disorder increases for high column density
sightlines. Alternatively, the intrinsic polarization efficiency of the dust
grain population might be lower for material along higher density sightlines.
We find no significant correlation between N and S. Comparison of observed
submillimeter polarization maps with synthetic polarization maps derived from
numerical simulations provides a promising method for testing star formation
theories. Realistic simulations should allow for the possibility of variable
intrinsic polarization efficiency. The measured levels of correlation among p,
N, and S provide points of comparison between observations and simulations.
