1. Miller Indices & Steriographic Projection
The Miller indices can be determined from the steriographic projection by measuring the angles relative to known crystallographic directions and applying the law-of-cosines.
(Figure 2-39 Cullity)
For r, s, and t to represent the angles between the normal of a plane and the a1, a2, and a3 axes respectively, then:
Where a, b, and c are the unit cell dimensions, and a/h, b/k, and c/l are the plane intercepts with the axes. The inner planar spacing, d, is equal to the distance between the origin and the plane (along a direction normal to the plane).

2. Vector Operations
Dot product:
Cross product: b a a
Volume:

3. Reciprocal Lattice Unit cell: a1, a2, a3
Reciprocal lattice unit cell: b1, b2, b3 defined by: b3 P B a3 C a2 O A a1

4. Reciprocal Lattice
Like the real-space lattice, the reciprocal space lattice also has a translation vector, Hhkl:
Where the length of Hhkl is equal to the reciprocal of the spacing of the (hkl) planes
Consider planes of a zone (i.e..: 2D reciprocal lattice).
Next overhead and (Figures A1-4, and A1-5 Cullity)

5. Zone Axis Planes could be translated so as not to
For planes of a zone, all plane normals lie in a common plane which is perpendicular to the zone axis. For example, if the zone axis labeled above is parallel to the coordinate system z -axis, then all plane normals lie in the x-y plane.
Planes could be translated so as not to
intersect at a common point.

6. Reciprocal Lattice (Zone Axis) Zone axis = ua1 + va2 + wa3 f A B C
(hkl) H
dhkl N O
H·AB = 0
H·AC = 0
Therefore the plane must be perpendicular to H.
dhkl = (a1/h)cosf = (a1/h)·n = (a1/h)·(H/H) = (a1/h)·{(hb1 + kb2 + lb3)/H}
dhkl = 1/H
Also:
Inner planar spacing:
(Zone Axis)
Zone axis = ua1 + va2 + wa3
Angles between reciprocal
lattice vectors.