1. The problem statement, all variables and given/known data Given the shortest reciprocal lattice of fcc is (2PI/a)(+-x+-y+-z). How can i show that (111) is perpendicular to the shortest reciprocal lattice? And how to show that other planes such as (001),(010),(110) is not perpendicular to the shortest reciprocal lattice? 2. Relevant equations Is it the plane will perpendicular to all the 6 reciprocal vector if it perpendicular to anyone of the reciprocal vectors,said 2Pi/a (x-y+z)? 3. The attempt at a solution I can get the answer by imagination, but i want to know how to prove it mathematically. I think it should use dot products but don't know how.