Exchange anisotropy in ferromagnet/antiferromagnet (FM/AF) films is usually introduced along the cooling field or FM magnetization direction. Here we investigate the dependence of the exchange anisotropy, loop bifurcation, and reversal mechanism on the cooling field direction using vector magnetometry. Three types of samples (FM=Fe, Ni/AF=FeF2, MnF2) have been studied where the AF layer is epitaxial (110), twinned (110), and polycrystalline. With an epitaxial AF which has one spin axis, the cooling field orients the exchange field along the spin axis. Applying the cooling field perpendicular to the spin axis results in bifurcated loops, whose shape evolves with the cooling field geometry and strength. With a twinned AF where there are two orthogonal spin axes, the exchange field direction is along the bisector of the spin axes that encompass the cooling field. With a polycrystalline AF, the exchange field direction is the same as the cooling field. Transverse hysteresis loops show that when the exchange field has a component perpendicular to the applied field, the magnetization reversal occurs by rotation in the direction of the perpendicular component. Our results demonstrate that in fluoride films, the exchange field is established primarily by the AF anisotropy direction, and only to a lesser extent the cooling field or the magnetization direction. The bifurcated loops are due to a distribution of AF anisotropies and large AF domain sizes. Furthermore, the magnetization reversal process is extremely sensitive to the exchange field direction.