"Orthonormal" means TWO things:
1) the vectors are orthogonal: their dot products are all 0. You checked that.
2) The length of each vector is 1. I don't see where you checked that.
For the other problem, are the two vectors orthogonal? Is their dot product 0? Is so are they of length 1? If not you can change a vectors length to 1 by dividing it by its length.