  We outline a general methodology to infer the inductive velocity field vector
in solar active regions. For the first time, both the field-aligned and the
cross-field velocity components are reconstructed. The cross-field velocity
solution accounts for the changes of the vertical magnetic field seen between a
pair of successive active-region vector magnetograms via the ideal induction
equation. The field-aligned velocity is obtained using the Doppler velocity and
the calculated cross-field velocity. Solving the ideal induction equation in
vector magnetograms measured at a given altitude in the solar atmosphere is an
under-determined problem. In response, our general formalism allows the use of
any additional constraint for the inductive cross-field velocity to enforce a
unique solution in the induction equation. As a result, our methodology can
give rise to new velocity solutions besides the one presented here. To
constrain the induction equation, we use a special case of the minimum
structure approximation that was introduced in previous studies and is already
employed here to resolve the $180^o$-ambiguity in the input vector
magnetograms. We reconstruct the inductive velocity for three active regions,
including NOAA AR 8210 for which previous results exist. Our solution
believably reproduces the horizontal flow patterns in the studied active
regions but breaks down in cases of localized rapid magnetic flux emergence or
submergence. Alternative approximations and constraints are possible and can be
accommodated into our general formalism.
