Finding a vector normal to a face of a polytope Suppose I have an $n-1$ dimensional facet of an $n$ dimensional polytope, where the facet is expressed by a set of points. For a given $n-2$ face of the facet, how can I find a vector that is both: normal to the face as well as being in the same hyperplane that the facet rests in?

To find a vector normal to $n-1$ given vectors, you can use the generalized cross product.

First find a vector normal to the facet from the generalized cross product of $n-1$ (non-coplanar) points defining the facet. Then find a vector normal to both the face and the facet normal from the generalized cross product of the facet normal and $n-2$ (non-coplanar) points defining the face.