text: For an infinite three-dimensional lattice , defined by its primitive vectors  and the subscript of integers , its reciprocal lattice  with the integer subscript  can be determined by generating its three reciprocal primitive vectors 
  where  is the scalar triple product. The choice of these  is to satisfy  as the known condition (There may be other condition.) of primitive translation vectors for the reciprocal lattice derived in the heuristic approach above and the section multi-dimensional Fourier series. This choice also satisfies the requirement of the reciprocal lattice  mathematically derived above. Using column vector representation of (reciprocal) primitive vectors, the formulae above can be rewritten using matrix inversion:
  This method appeals to the definition, and allows generalization to arbitrary dimensions. The cross product formula dominates introductory materials on crystallography.
  The above definition is called the "physics" definition, as the factor of  comes naturally from the study of periodic structures. An essentially equivalent definition, the "crystallographer's" definition, comes from defining the reciprocal lattice . which changes the reciprocal primitive vectors to be
  and so on for the other primitive vectors. The crystallographer's definition has the advantage that the definition of  is just the reciprocal magnitude of  in the direction of , dropping the factor of . This can simplify certain mathematical manipulations, and expresses reciprocal lattice dimensions in units of spatial frequency. It is a matter of taste which definition of the lattice is used, as long as the two are not mixed.
   is conventionally written as  or , called Miller indices;  is replaced with ,  replaced with , and  replaced with . Each lattice point  in the reciprocal lattice corresponds to a set of lattice planes  in the real space lattice. (A lattice plane is a plane crossing lattice points.) The direction of the reciprocal lattice vector corresponds to the normal to the real space planes. The magnitude of the reciprocal lattice vector  is given in reciprocal length and is equal to the reciprocal of the interplanar spacing of the real space planes.