  In this paper, we determine the Hausdorff dimension of the set of points with
divergent trajectories on the product of certain homogeneous spaces. The flow
is allowed to be weighted with respect to the factors in the product space. The
result is derived from its counterpart in Diophantine approximation. In doing
this, we introduce a notion of jointly singular matrix tuples, and extend the
dimension formula for singular matrices to such matrix tuples.
