  On the basis of Eilenberger theory, we study the vortex state when a magnetic
field is applied nearly parallel to the ab plane in a chiral p-wave
superconductor with a large anisotropy ratio of ab and c, as in Sr2RuO4. We
quantitatively estimate the field dependence of the pair potential,
magnetization, and flux line lattice form factor, and study the transition from
the chiral p_- state at low fields to the nonchiral p_y state at high fields.
Even for exactly parallel fields to the ab plane, transverse fields exist in
the chiral state. The chiral-nonchiral transition disappears when the magnetic
field orientation is tilted within 1 degree from the ab plane. This may be a
reason why the experimental detection of this transition is difficult.
