Given a sequence of positive integers p=(p1,...,pn), let Sp
denote the family of all sequences of positive integers x=(x1,...,xn)
such that xi≤pi for all i. Two families of sequences (or vectors),
A,B⊆Sp, are said to be r-cross-intersecting if no matter how we
select x∈A and y∈B, there are at least r distinct indices i
such that xi=yi. We determine the maximum value of ∣A∣⋅∣B∣ over all
pairs of r- cross-intersecting families and characterize the extremal pairs
for r≥1, provided that minpi>r+1. The case minpi≤r+1 is
quite different. For this case, we have a conjecture, which we can verify under
additional assumptions. Our results generalize and strengthen several previous
results by Berge, Frankl, F\"uredi, Livingston, Moon, and Tokushige, and
answers a question of Zhang