Chapter 12.4, Problem 54E

If v₁, v₂, and v₃ are noncoplanar vectors, let

${\text{k}}_{\text{1}}\text{=}\frac{{\text{v}}_{\text{2}}\text{×}{\text{v}}_{\text{3}}}{{\text{v}}_{\text{1}}\cdot \left({\text{v}}_{\text{2}}\text{×}{\text{v}}_{\text{3}}\right)}$ ${\text{k}}_{\text{2}}\text{=}\frac{{\text{v}}_{\text{3}}\text{×}{\text{v}}_{\text{1}}}{{\text{v}}_{\text{1}}\cdot \left({\text{v}}_{\text{2}}\text{×}{\text{v}}_{\text{3}}\right)}$

${\text{k}}_{\text{3}}\text{=}\frac{{\text{v}}_{\text{1}}\text{×}{\text{v}}_{\text{2}}}{{\text{v}}_{\text{1}}\cdot \left({\text{v}}_{\text{2}}\text{×}{\text{v}}_{\text{3}}\right)}$

(These vectors occur in the study of crystallography. Vectors of the form n₁ v₁ + n₂ v₂ + n₃ v₃, 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.

(c) Show that ${\text{k}}_{\text{1}}\text{×}\left({\text{k}}_{\text{2}}\text{×}{\text{k}}_{\text{3}}\right)\text{=}\frac{\text{1}}{{\text{v}}_{\text{1}}\cdot \left({\text{v}}_{\text{2}}\text{×}{\text{v}}_{\text{3}}\right)}$