A concise mathematical description of the relationship between the real and reciprocal lattice vectors is given by:

Where V is the volume of the real lattice cell and is given vectorially by: V= a . b Ù c

These equations show that:	a* is perpendicular to both b and c
	b* is perpendicular to both c and a
	c* is perpendicular to both a and b

These equations also enable you to calculate the length of the reciprocal vectors.

However, same relationships can be expressed in other ways and the following vector relationships are also true:

	a* . b = a* . c = 0	i.e. a* is perpendicular to both b and c
	b* . a = b* . c = 0	i.e. b* is perpendicular to both c and a
	c* . a = c* . b = 0	i.e. c* is perpendicular to both a and b

Another way of stating the relationships is:

These equations show that:	a* is perpendicular to the (100) plane
	b* is perpendicular to the (010) plane
	c* is perpendicular to the (001) plane

Note: the reciprocal vector a* is always perpendicular to the plane (100) (since the (100) plane contains the b and c vectors) but the real lattice vector a might not be, depending on the crystal system. The vector a is always perpendicular to the (100) for the cubic system.

It is also true that : a* . a = b* . b = c* . c = 1

These equations give the magnitudes of the reciprocal vectors. Note that |a*| = 1 / |a| only for orthogonal lattices (e.g. cubic, tetragonal and orthorhombic).

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