Proof of Conditions For Sets of Vectors To Be Reciprocal

Theorem

Two sets of vectorsandare said to be reciprocal if and only if

Proof

Sets of vectorsandare reciprocal ifand (1)

From the last equation we concude thatis orthogonal toandso is in the same direction asand we can write

Taking the dot product of this on the left bygives

The same argument givesand

Alsoand similarly forand

Finally,sinceis perpendicular toand similarly for all the dot products in (1).