For an infinite three-dimensional lattice R n = n 1 a 1 + n 2 a 2 + n 3 a 3 , defined by its primitive vectors ( a 1 , a 2 , a 3 ) and the subscript of integers n = ( n 1 , n 2 , n 3 ) , its reciprocal lattice G m = m 1 b 1 + m 2 b 2 + m 3 b 3 with the integer subscript m = ( m 1 , m 2 , m 3 ) can be determined by generating its three reciprocal primitive vectors ( b 1 , b 2 , b 3 ) b 1 = 2 π π V a 2 × × a 3 b 2 = 2 π π V a 3 × × a 1 b 3 = 2 π π V a 1 × × a 2 where V = a 1 ⋅ ⋅ ( a 2 × × a 3 ) = a 2 ⋅ ⋅ ( a 3 × × a 1 ) = a 3 ⋅ ⋅ ( a 1 × × a 2 ) is the scalar triple product. The choice of these ( b 1 , b 2 , b 3 ) is to satisfy a i ⋅ ⋅ b j = 2 π π δ δ i j 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 e i G m ⋅ ⋅ R n = 1 mathematically derived above. Using column vector representation of (reciprocal) primitive vectors, the formulae above can be rewritten using matrix inversion :