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

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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 ni 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 ki is perpendicular to vj if i ≠ j.
(b) Show that ki ? vi = 1 for i = 1, 2, 3.

(c) Show that k1 • (k2 x k3) − 1/v1 • (v2 x v3).

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