Real Space Lattice Vectors

\(\vec a\)= \(\vec e_x+\) \(\vec e_y+\) \(\vec e_z\)
\(\vec b\)= \(\vec e_x+\) \(\vec e_y+\) \(\vec e_z\)
\(\vec c\)= \(\vec e_x+\) \(\vec e_y+\) \(\vec e_z\)

Reciprocal Space Lattice Vectors

\(\vec {a'}\)= \(\vec e_x+\) \(\vec e_y+\) \(\vec e_z\)
\(\vec {b'}\)= \(\vec e_x+\) \(\vec e_y+\) \(\vec e_z\)
\(\vec {c'}\)= \(\vec e_x+\) \(\vec e_y+\) \(\vec e_z\)

Formulas

\[\vec a=\frac{\vec{b'} \times \vec {c'}}{V'} \quad\vec b=\frac{\vec{ c'} \times \vec {a'}}{V'}\quad \vec c=\frac{\vec{ a'} \times \vec {b'}}{V'}\quad V'=\vec{a'} \cdot (\vec{b'} \times \vec{c'})\]

\[\vec {a'}=\frac{\vec{b} \times \vec {c}}{V}\quad \vec {b'}=\frac{\vec{ c} \times \vec {a}}{V}\quad \vec {c'}=\frac{\vec{ a} \times \vec {b}}{V}\quad V=\vec{a} \cdot (\vec{b} \times \vec{c})\]