ux(i-3k) is a vector- it has i, j, and k components. Setting the i component equal to 0, the j component equal to 2 and the k component equal to 0 gives you three equations for the three coefficients of i, j, and k in u.
If the answer has a parameter (t) in it, then you don't need another equation. Just get c in terms of a, say, and substitute into u = ai + ck.
Otherwise, (and I haven't checked whether the equation is different) equate the magnitude of each side of (ai + ck)(i-3k)=2j (I assume you know how to find the magnitude of a vector).