What can the answer be? 2. Reciprocal basis and dual vectors

Abstract

We usually express vectors as a sum of basis vectors which are mutually perpendicular; of unit length. In some situations; such as the description of crystals; it is necessary to use basis vectors which have any length; any angle between them. Solving for the coefficients in such an expansion introduces the concept of reciprocal vectors or dual vectors.They are the natural language to use in describing phenomena periodic in space; such as waves; crystal lattices. Generalisation of this concept to infinite dimensions leads to Dirac's notation for quantum states.