A set of vertices W resolves a graph G if every vertex of G is uniquely determined by its vector of distances to the vertices in W. The metric dimension for G, denoted by dim(G), is the minimum cardinality of a resolving set of G. In order to study the metric dimension for the hierarchical product Gu22  Gu11 of two rooted graphs Gu22 and Gu11, we first introduce a new parameter, the rooted metric dimension rdim(Gu11) for a rooted graph Gu11. If G1 is not a path with an end-vertex u1, we show that dim(Gu22  Gu11) = |V(G2)|• rdim(Gu11), where |V(G2)| is the order of G2. If G1 is a path with an end-vertex u1, we obtain some tight inequalities for dim(Gu22  Gu11). Finally, we show that similar results hold for the fractional metric dimension.