If v 1 , v 2 , and v 3 are non coplanar vectors, let (These vectors occur in the study of crystallography Vectors of the form n 1 v 1 n 2 v 2 n 3 v 3 , where each n i is an integer, form a lattice for a crystal Vectors written similarly in terms of k 1 , k 2 , and k 3 form the reciprocal lattice ) ...

a k is perpendicular to v if i j by the definition of k and Th ...

If v 1 , v 2 , and v 3 are non-coplanar vectors, let (These vectors occur

Question:

If v₁, v₂, and v₃ are non-coplanar vectors, let

Vз X VI V1· (v2 X V3) V2 X V3 ki k2 V1 • (V2 × V3) Vị X V2 k3 (v2 × V3) K3 Vị •

(These vectors occur in the study of crystallography. Vectors of the form n₁v₁ + n₂v₂ + n₃v₃, where each n_i is an integer, form a lattice for a crystal. Vectors written similarly in terms of k₁, k₂, and k₃ form the reciprocal lattice.)
(a) Show that k_i is perpendicular to v_j if i ≠ j.
(b) Show that k_i ? v_i = 1 for i = 1, 2, 3.