Deduce primitive lattice vectors from position vector.

maqdah

1. The problem statement, all variables and given/known data
given the following position vector:

R = (10n1 + 9n2 + 19n3)(a/10) x + 6(n2+n3)(a/5) y + 2(n3)a z

where n1, n2 and n3 are integers
Find the primitive lattice vectors.

2. Relevant equations
any position vector of a lattice point is of the type
R= c1 a1 + c2 a2 + c3 a3;
and a position vector like the one showed above is a linear combination of the primitive lattice vectors a1, a2 and a3.


3. The attempt at a solution

I think I solved the question correctly, but my intuition tells me its wrong:

we can do the following:
a1 = a/10 X
a2 = a/5 Y
a3 = a Z

In our case, since n1,n2 and n3 are just integers:

c1 = 10n1 + 9n2 + 19n3
c2 = 6(n2 + n3)
c3 = 2n3

maqdah

Any hint in the right direction is helpful?