  To any $k$-dimensional subspace of $\mathbb Q^n$ one can naturally associate
a point in the Grassmannian ${\rm Gr}_{n,k}(\mathbb R)$ and two shapes of
lattices of rank $k$ and $n-k$ respectively. These lattices originate by
intersecting the $k$-dimensional subspace with the lattice $\mathbb Z^n$. Using
unipotent dynamics we prove simultaneous equidistribution of all of these
objects under a congruence conditions when $(k,n) \neq (2,4)$.
