PX153 - A7 - reciprocal vectors

relevant for non-orthogonal basis sets
- eg: we will use it to determine components of a vector in a non-orthogonal basis set
  - $\vec{a} = \sum a_{i} {\vec{e}}_{i}$
    where, $a_{i} =$ components ; ${\vec{e}}_{i} =$ basis vectors
  - for cartesian: $\vec{e} \cdot {\vec{e}}_{j} = δ_{i j}$ and, $\vec{a} \cdot \underset{―}{{\hat{e}}_{x}} = {\vec{a}}_{x}$
    - easy to find components
  - but in non-orthogonal:
    $\vec{e} \cdot {\vec{e}}_{j} \neq δ_{i j}$
    and, ${\vec{e}}_{i} \cdot \vec{a} \neq a_{i}$
    $\vec{a} = \sum_{i} a_{i} {\vec{e}}_{i}$
    - not easy to find components
define reciprocal vectors:

\begin{matrix} e_{1}^{'} = \frac{e_{2} \times e_{3}}{e_{1} \cdot (e_{2} \times e_{3})} \\ e_{2}^{'} = \frac{e_{3} \times e_{1}}{e_{1} \cdot (e_{2} \times e_{3})} \\ e_{3}^{'} = \frac{e_{1} \times e_{2}}{e_{1} \cdot (e_{2} \times e_{3})} \end{matrix}

${\vec{e}}_{i} \cdot {\vec{e}}_{j}^{'} = δ_{i j}$
$a_{i} = {\vec{e}}_{i}^{'} \cdot a$