(a) Given a Bravais lattice, let a1 be a vector joining a particular point P to one of its nearest neighbors. Let P′ be a lattice point not on the line through P in the direction of a1 that is as close to the line as any other lattice point, and let a2 join P to P′. Let P′′ be a lattice point not on the plane through P determined by a1 and a2 that is as close to the plane as any other lattice point, and let a3 join P to P′′. Prove that a1, a2, and a3 are a set of primitive vectors for the Bravais lattice.
(b) Prove that a Bravais lattice can be defined as a discrete set of vectors, not all in a plane, closed under addition and subtraction (as described on page 70).