1. If `vec(a),vec(b),vec(c)` and `vec(a'),vec(b'),vec(c')` are 2 sets of non coplanar vectors such that `vec(a)*vec(a') =vec(b)*vec(b')=vec(c)*vec(c')=1` and `vec(a)*vec(b) =vec(a)*vec(c) =vec(b)*vec(a)=vec(b)*vec(c)=vec(c)*vec(a)=vec(c)*vec(b)=0` then

`vec(a),vec(b),vec(c)` and `vec(a'),vec(b'),vec(c')` are said to be constitute a reciprocal system of vectors.

2. Reciprocal system of vectors exists only in case of dot product.

3. It is possible to define `vec(a'),vec(b'),vec(c')` in terms of `vec(a),vec(b),vec(c)` as.

`vec(a') = (vec(b) xx vec(c))/[vec(a)vec(b)vec(c) ] ; vec(b') = (vec(c) xx vec(a))/[vec(a)vec(b)vec(c) ] ;vec(c') = (vec(a) xx vec(b))/[vec(a)vec(b)vec(c) ] ` ( `[vec(a)vec(b)vec(c) ] ne 0`)

Note:
(i) `vec(a) xx vec(a') +vec(b) xx vec(b')+vec(c) xx vec(c') =0` i.e., `( vec(a)xx (vec(b) xx vec(c))+vec(b) xx(vec(c) xx vec(a)) + vec(c) xx (vec(a) xx vec(b)) ) /[ vec(a)vec(b)vec(c)]`

(ii) `(vec(a)+vec(b)+vec(c)) * (vec(a')+vec(b')+vec(c')) =3` (as `hat(a)*hat(b')=hat(a)*hat(c') = 0` etc)

(iii) If `[vec(a)vec(b)vec(c)] =V` then ` [vec(a')vec(b')vec(c')] =1/V => [vec(a)vec(b)vec(c) ] [vec(a')vec(b' )vec(c') ] =1`

(iv) `vec(a') xx vec(b')+vec(b') xx vec(c')+vec(c') xx vec(a') = (vec(a)+vec(b)+vec(c))/ [ vec(a)vec(b)vec(c)] , [vec(a)vec(b)vec(c)] ne 0`