Proof That Any Vector Can Be Written In Terms Of Reciprocal Vectors

Theorem

Given sets of reciprocal vectorsandthen any vectorcan be written in terms of

Proof

Use the identityand rearrange it to givevec u cdot (vec v times vec w ) =vec w cdot (vec u times vec v )= vec v cdot (vec u times vec v ) hence(1)

From the definitions of vec a' , vec b' , vec c',we can write (1) as