Finding values in a space that matches a relational query
One approach is to guess $a$ (i.e., iterate over all possibilities for it, and the following for each possibility).  Then, you can infer the value of $b$, namely, $b = a r_b/r_a$, so test whether $b \in B$.  Do the same for each of $c,\dots,z$.  If they all work, then you have found a solution; otherwise, proceed on to your next guess at $a$.  This should give you approximately a linear running time if you use the right data structures for storing the vectors (e.g., a hashtable).
A similar approach is to compute the vectors $A/r_a$, $B/r_b$, ..., $Z/r_z$, and then look for an item that occurs in all of those scaled vectors, e.g., using a hashtable or a counter or by sorting and merging the sorted lists.