If , , and are noncoplanar vectors, let (These vectors
Chapter 10, Problem 10.5.1(choose chapter or problem)
If , , and are noncoplanar vectors, let (These vectors occur in the study of crystallography. Vectors of the form , where each is an integer, form a lattice for a crystal. Vectors written similarly in terms of , , and form the reciprocal lattice.) (a) Show that is perpendicular to if . (b) Show that for . (c) Show that . k1 k2 k3 1 v1 v2 v3 ki vi (f ) Two lines perpendicular to a plane are parallel (g) Two planes parallel to a line are parallel. (h) Two planes perpendicular to a line are parallel. (i) Two planes either intersect or are parallel. ( j) Two lines either intersect or are parallel. (k) A plane and a line either intersect or are parallel.
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