Subsection 1.1.8 Reciprocal Vectors
If the three vectors \(\vec{a}\text{,}\) \(\vec{b}\text{,}\) and \(\vec{c}\) are non-coplanar, i.e., \([\vec{a}\vec{b}\vec{c}] \neq 0,\) and the vectors \(\vec{a'}\text{,}\) \(\vec{b'}\text{,}\) and \(\vec{c'}\) are defind as
\begin{equation*}
\vec{a'} = \frac{\vec{b}\times\vec{c}}{[\vec{a}\vec{b}\vec{c}]}; \quad \vec{b'}
= \frac{\vec{c}\times\vec{a}}{[\vec{a}\vec{b}\vec{c}]}; \quad \text{and}\quad \vec{c'}
= \frac{\vec{a}\times\vec{b}}{[\vec{a}\vec{b}\vec{c}]}
\end{equation*}
then the vectors \(\vec{a}\text{,}\) \(\vec{b}\text{,}\) \(\vec{c}\) and \(\vec{a'}\text{,}\) \(\vec{b'}\text{,}\) \(\vec{c'}\) are called the reciprocal sets of vectors.
