  Based on the two-dimensional mean-field equations for pancake-shaped dipolar
Bose-Einstein condensates in a rotating frame with both attractive and
repulsive dipole-dipole interaction (DDI) as well as arbitrary polarization
angle, we study the profiles of the single vortex state and show how the
critical rotational frequency change with the s-wave contact interaction
strengths, DDI strengths and the polarization angles. In addition, we find
numerically that at the `magic angle' $\vartheta=\arccos(\sqrt{3}/3)$, the
critical rotational frequency is almost independent of the DDI strength. By
numerically solving the dipolar GPE at high rotational speed, we identify
different patterns of vortex lattices which strongly depend on the polarization
direction. As a result, we undergo a study of vortex lattice structures for the
whole regime of polarization direction and find evidence that the vortex
lattice orientation tends to be aligned with the direction of the dipoles.
