Fractional coordinates

Let us consider a system of periodic structure in space and use ${\displaystyle {\mathbf {a} }}$ , ${\displaystyle \mathbf {b} }$, and ${\displaystyle \mathbf {c} }$ as the three independent period vectors, forming a right-handed triad, which are also the edge vectors of a cell of the system. Then any vector ${\displaystyle \mathbf {r} }$ in Cartesian coordiantes can be written as a linear combination of the period vectors

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