Modulus of position vector

Arnold, H.

doi:10.1107/97809553602060000511

International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by Th. Hahn

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International Tables for Crystallography (2006). Vol. A. ch. 5.2, p. 87

Section 5.2.2.2. Modulus of position vector

H. Arnold^a ^‡

^a Institut für Kristallographie, Rheinisch-Westfälische Technische Hochschule, Aachen, Germany

5.2.2.2. Modulus of position vector

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The modulus r of the position vector r gives the distance of the point x, y, z from the origin. Its square is obtained by the scalar product $[\eqalignno{{\bf r}^{\;{\bi t}} \cdot {\bf r} = r^{2} &= (x,y,z) \pmatrix{{\bf a}\cr {\bf b}\cr {\bf c}\cr} ({\bf a},{\bf b},{\bf c}) \pmatrix{x\cr y\cr z\cr}\cr &= (x,y,z) {\bi G} \pmatrix{x\cr y\cr z\cr}\cr &= x^{2}a^{2} + y^{2}b^{2} + z^{2}c^{2} + 2yzbc \cos \alpha &\cr &\quad + 2xzac \cos \beta + 2xyab \cos \gamma,}]$ with $[{\bf r}^{\;\bi t}]$ the transposed representation of r; a, b, c the moduli of the basis vectors a, b, c (lattice parameters); G the metric matrix of direct space; and α, β, γ the angles of the unit cell.

The same considerations apply to the vector $[{\bf r}^{*}]$ in reciprocal space and its modulus $[r^{*}]$ . Here, $[{\bi G}^{*}]$ is applied. Note that $[{\bf r}^{*}]$ and $[r^{*}]$ are independent of the choice of the origin in direct space.

References

International Tables for Crystallography (2006). Vol. A. ch. 5.2, p. 87