Given non-coplanar vectors a, b, c (with [a b c] != 0), the reciprocal vectors are defined as: a' = (b x c)/[a b c], b' = (c x a)/[a b c], c' = (a x b)/[a b c].

Orthonormality-like properties: a.a'=1, b.b'=1, c.c'=1 (each vector dot its reciprocal gives 1). a.b'=0, a.c'=0, b.a'=0, b.c'=0, c.a'=0, c.b'=0 (each vector dot a different reciprocal gives 0).

The reciprocal of the reciprocal system is the original system.

STP of reciprocal vectors: [a' b' c'] = 1/[a b c].

Resolution of vectors: any vector r can be resolved along non-orthogonal a, b, c as r = (r.a')a + (r.b')b + (r.c')c. The reciprocal vectors provide the coefficients.

For orthogonal unit vectors, the reciprocal system equals the original: i'=i, j'=j, k'=k (since [i j k]=1 and j x k=i, etc.).

The reciprocal system appears occasionally in JEE Advanced. The typical question asks to find a.p+b.q+c.r where p=(b x c)/[a b c] etc. Since a.p=1, b.q=1, c.r=1, the answer is 3.

Physical application: in crystallography, reciprocal lattice vectors are exactly these reciprocal vectors. They define the Fourier space dual to the real-space lattice.