If $\vec{a},\vec{b},\vec{c}$ are 3 non-zero, non-collinear, non-coplanar vectors then the reciprocal system of vectors for $\vec{a},\vec{b},\vec{c}$ are $$ \vec{a}'=\frac{\vec{b}\times\vec{c}}{[\vec{a} \;\vec{b}\;\vec{c}]}\;;\vec{b}'=\frac{\vec{c}\times\vec{a}}{[\vec{a} \;\vec{b}\;\vec{c}]}\;;\vec{c}'=\frac{\vec{a}\times\vec{b}}{[\vec{a} \;\vec{b}\;\vec{c}]} $$
Mathematically reciprocal vectors are vectors in the direction perpendicular to the remaining two vectors. And $\vec{a}\cdot\vec{a}'=\vec{b}\cdot\vec{b}'=\vec{c}\cdot\vec{c}'=1$, so it seems like these are defined to introduce some kind of reciprocal to a vector since the division of vectors are undefined.
Could someone give more insight into why the concept of reciprocal vectors are introduced and what exactly is the purpose of such a concept ?
Is there a simple and basic physical or mathematical problem where these are made use of ?