There are four miller indices (hklj) for the hexagonal lattice, the third being redudant: l=-(h+k) (1) Given the basis vectors a1,a2,a3 I can certainly see that: a3=-(a1+a2) But how does this immidiatly lead me to the relation (1) between the miller indices?
Spontaneously, I don't know the answer, but I think it is relatively easy to work out. Have you tried? If you really encounter problems, we all are willing to help you.
The problem is how exactly to relate the miller indices given (-a1+a2)=a3. After all miller indices for a plane are obtained as inverses of the coordinates for the intersection of the lattice vectors with the plabe.
Oh, so these are the a1,a2,a3 in the system with four indices. You have and a4 as well. Right? Sorry, I was confused.
Hint: Miller indices refer to a point in reciprocal space, hence you have to construct the reciprocal space basis vectors. G = H a* + K b* + L c*(for the "normal" 3 Miller indices).