Theorem
Two sets of vectors
and
are said to be reciprocal if and only if
Proof
Sets of vectorsandare reciprocal if
and
(1)
From the last equation we concude that
is orthogonal to
and
so is in the same direction as
and we can write
Taking the dot product of this on the left by
gives
The same argument gives
and
Also
and similarly for
and
Finally,
sinceis perpendicular to
and similarly for all the dot products in (1).