International
Tables for Crystallography Volume C Mathematical, physical and chemical tables Edited by E. Prince © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. C, ch. 1.1, pp. 45

The angles between the normal of a crystal face and the basis vectors a, b, c are called the direction angles of that face. They may be calculated as angles between the corresponding reciprocallattice vector r* and the basis vectors , and : The three equations can be combined to give The first formula gives the ratios between a, b, and c, if for any face of the crystal the indices (hkl) and the direction angles λ, μ, and ν are known. Once the axial ratios are known, the indices of any other face can be obtained from its direction angles by using the second formula.
Similarly, the angles between a directlattice vector t and the reciprocal basis vectors , and are given by The angle between two directlattice vectors and or between two corresponding point rows and may be derived from the scalar product as Analogously, the angle between two reciprocallattice vectors and or between two corresponding point rows and or between the normals of two corresponding crystal faces and may be calculated as with
Finally, the angle between a first direction [uvw] of the direct lattice and a second direction [hkl] of the reciprocal lattice may also be derived from the scalar product of the corresponding vectors t and r*.