Vectors and vector coefficients

Wondratschek, H.

doi:10.1107/97809553602060000538

International
Tables for
Crystallography
Volume A1
Symmetry relations between space groups
Edited by Hans Wondratschek and Ulrich Müller

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International Tables for Crystallography (2006). Vol. A1. ch. 1.2, pp. 8-9 | 1 | 2 |

Section 1.2.2.6. Vectors and vector coefficients

Hans Wondratschek^a ^*

^a Institut für Kristallographie, Universität, D-76128 Karlsruhe, Germany
Correspondence e-mail: wondra@physik.uni-karlsruhe.de

1.2.2.6. Vectors and vector coefficients

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In crystallography, vectors and their coefficients as well as points and their coordinates are used for the description of crystal structures. Vectors represent translation shifts, distance and Patterson vectors, reciprocal-lattice vectors etc. With respect to a given basis a vector has three coefficients. In contrast to the coordinates of a point, these coefficients do not change if the origin of the coordinate system is shifted. In the usual description by columns, the vector coefficients cannot be distinguished from the point coordinates, but in the augmented-column description the difference becomes visible: the vector from the point P to the point Q has the coefficients [v_1=q_1-p_1] , [v_2=q_2-p_2] , [v_3=q_3-p_3] , [1 - 1] . Thus, the column of the coefficients of a vector is not augmented by `1' but by `0'. Therefore, when the point P is mapped onto the point $[\tilde{P}]$ by $[\tilde{{\bi x}}= {\bi W}\,{\bi x} + {\bi w}]$ according to equation (1.2.2.3), then the vector $[{\bf v} = \overrightarrow{P\,Q}]$ is mapped onto the vector $[ \tilde{\bf{v}} = \overrightarrow{\tilde{P}\,\tilde{Q}}]$ by transforming its coefficients by $[\tilde{{\bi v}}={\bi W}\,{\bi v}]$ , because the coefficients [w_j] are multiplied by the number `0' augmenting the column $[{\bi v} = (v_j)]$ . Indeed, the distance vector $[{\bf v} = \overrightarrow{P\,Q}]$ is not changed when the whole space is mapped onto itself by a translation.

Remarks :

(1) The difference in transformation behaviour between the point coordinates x and the vector coefficients v is not visible in the equations where the symbols $[\specialfonts{\bbsf x}]$ and $[\specialfonts{\bbsf v}]$ are used, but is obvious only if the columns are written in full, viz $[ \left (\matrix{x_1 \cr x_2 \cr x_3\cr \noalign{\vskip.3em} \noalign {\hrule} \cr 1}\right) {\rm\,\,\, and\,\,\, } \left (\matrix{v_1 \cr v_2 \cr v_3\cr \noalign{\vskip.3em} \noalign {\hrule} \cr 0} \right).]$
(2) The transformation behaviour of the vector coefficients is also apparent if the vector is understood to be a translation vector and the transformation behaviour of the translation is considered as in the last paragraph of the next section.
(3) The transformation $[\tilde{{\bi v}}={\bi W}\,{\bi v}]$ is called an orthogonal mapping if W is the matrix part of an isometry.

References

International Tables for Crystallography (2006). Vol. A1. ch. 1.2, pp. 8-9