  In this work we investigate 1+2 reactions within the framework of the
hyperspherical adiabatic expansion method. To this aim two integral relations,
derived from the Kohn variational principle, are used. A detailed derivation of
these relations is shown. The expressions derived are general, not restricted
to relative $s$ partial waves, and with applicability in multichannel
reactions. The convergence of the ${\cal K}$-matrix in terms of the adiabatic
potentials is investigated. Together with a simple model case used as a test
for the method, we show results for the collision of a $^4$He atom on a \dimer
dimer (only the elastic channel open), and for collisions involving a $^6$Li
and two $^4$He atoms (two channels open).
