Given non-coplanar vectors a, b, c ([a b c] != 0), the reciprocal system consists of: a' = (b x c)/[a b c], b' = (c x a)/[a b c], c' = (a x b)/[a b c].

Properties:

• a.a' = 1, b.b' = 1, c.c' = 1.
• a.b' = 0, a.c' = 0, b.a' = 0, b.c' = 0, c.a' = 0, c.b' = 0.
• [a' b' c'] = 1/[a b c].
• The reciprocal of the reciprocal system gives back the original.

Use in JEE: Any vector r can be resolved along non-orthogonal directions a, b, c as r = (r.a')a + (r.b')b + (r.c')c. Wait, that's not right. The correct resolution is: r = (r.a')a + (r.b')b + (r.c')c only if a', b', c' are the reciprocal vectors. Actually: r = (r.a')a + (r.b')b + (r.c')c is incorrect. The correct form is r = (r.a)a' + (r.b)b' + (r.c)c'... No. The correct resolution: if r = xa+yb+zc, then x = r.a', y = r.b', z = r.c'. So r = (r.a')a + (r.b')b + (r.c')c.