Select a subset of the columns in $3\times n$ matrix, is it NP-hard?
Your problem is NP-complete, indeed. You can, e.g., reduce PARTITION to your problem. If you want to know if there is a subset of the set $\{d_1,\ldots,d_k\}$ that sums up to exactly $D := \frac{1}{2} \sum_{i=1}^k d_i$, you can use your problem by setting $b_i := 1$ and $a_i := d_i$ for all $i$ and by setting $c_1 := D$, $c_2 = 0$, and $c_3 := D$ as well as $b := n-1$.