  We present a three-dimensional (3-D) dynamical model for the broad [Fe III]
emission observed in Eta Carinae using the Hubble Space Telescope/Space
Telescope Imaging Spectrograph (HST/STIS). This model is based on full 3-D
Smoothed Particle Hydrodynamics (SPH) simulations of Eta Car's binary colliding
winds. Radiative transfer codes are used to generate synthetic spectro-images
of [Fe III] emission line structures at various observed orbital phases and
STIS slit position angles (PAs). Through a parameter study that varies the
orbital inclination i, the PA {\theta} that the orbital plane projection of the
line-of-sight makes with the apastron side of the semi-major axis, and the PA
on the sky of the orbital axis, we are able, for the first time, to tightly
constrain the absolute 3-D orientation of the binary orbit. To simultaneously
reproduce the blue-shifted emission arcs observed at orbital phase 0.976, STIS
slit PA = +38 degrees, and the temporal variations in emission seen at negative
slit PAs, the binary needs to have an i \approx 130 to 145 degrees, {\theta}
\approx -15 to +30 degrees, and an orbital axis projected on the sky at a PA
\approx 302 to 327 degrees east of north. This represents a system with an
orbital axis that is closely aligned with the inferred polar axis of the
Homunculus nebula, in 3-D. The companion star, Eta B, thus orbits clockwise on
the sky and is on the observer's side of the system at apastron. This
orientation has important implications for theories for the formation of the
Homunculus and helps lay the groundwork for orbital modeling to determine the
stellar masses.
