flat similarity

[gusl]

I previously inquired about a natural measure of "hyperplane similarity". But what I meant was "flat similarity": hyperplanes are flats with codimension 1, which makes this easy (we can simply take the dot product of the unit normals)

So again:
(1) What is a natural measure of flat similarity? Let's call our flats V and W. WOLOG, let's assume that V and W go through the origin.

rdore proposed that projecting to the orthogonal complement of V ∩ W won't change the answer. However, in general this won't give you vectors. Can we instead pick an arbitrary plane that is orthogonal to both flats (i.e. a 2D subspace of the above), thus ensuring that the projections are vectors?

So here's another basic question:

(2) Given two flats that go through the origin (with dimension m and n), what is the (a) maximum, and the (b) minimum/typical dimension of their intersection?

Thanks to Don Sheehy, I learned that the space of k-flats forms a manifold known as the Grassmannian, in which we can talk about geodesics. Geodesics give at least a partial order, which may be a first step towards a metric.

I'm feeling generally shocked at my ignorance of geometry. Here are some concepts to study:

http://en.wikipedia.org/wiki/Plücker_coordinates
http://en.wikipedia.org/wiki/Vector_bundle
http://en.wikipedia.org/wiki/Gauss_map
http://en.wikipedia.org/wiki/Gaussian_curvature
http://en.wikipedia.org/wiki/Real_projective_plane