... non-coplanar vectors a 1, a 2, and a 3that leave all theproperties of the crystal unchanged after the shift as a whole by any of those vectors. As a result, any lattice point Rcould be obtained ... not parallel to a 1, a 3as the shortest period not coplanar to a 1and a 2.• Vectors a ichosen in such a way are called primitive.• The volume cell enclosed by the primitive vectors is called ... (a 1 [a 2 a 3]) = (a 3[ a 1 a 2]) = (a 2 [a 3 a 1]).1.2. Show that only n = 1, 2, 3, 6 are available.1.3. We have mentioned that primitive vectors are not unique. New vectors can be definedas a i=kβik a k,the...