dimensionless and coordinate-free?

A lot of talk about Buckingham pi in my world. This is a theorem that says that any dimensional equation in physics with k dimensional inputs can be re-written as a dimensionless equation with fewer than k dimensionless inputs. But this is useless when we think about geometric equations—and many equations in physics are geometric.

Consider, for example, the coordinate-free equation F=ma. This equation has two dimensional vector terms. If we apply Buckingham pi, we get three coupled equations with non-scalar, non-coordinate-free dimensionless ratios. That's terrible, and useless! Can we replace Buckingham pi with something that makes equations that are both dimensionless and coordinate-free?