  An amplitude analysis of $\Lambda^+_c \to pK^-\pi^+$ decays together with a
measurement of the $\Lambda^+_c$ polarization vector in semileptonic beauty
hadron decays is presented. A sample of $400\,000$ candidates is selected from
proton-proton collisions recorded by the LHCb detector at a center-of-mass
energy of 13 TeV. An amplitude model is developed and the resonance fractions
as well as two- and three-body decay parameters are reported. The mass and
width of the $\Lambda(2000)$ state are also determined. A significant
$\Lambda^+_c$ polarization is found. A large sensitivity of the $\Lambda^+_c
\to pK^-\pi^+$ decay to the polarization is seen, making the amplitude model
suitable for $\Lambda^+_c$ polarization measurements in other systems.
