Scalar product

Sands, D. E.

doi:10.1107/97809553602060000559

International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shumeli

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International Tables for Crystallography (2006). Vol. B. ch. 3.1, p. 348 | 1 | 2 |

Section 3.1.2. Scalar product

D. E. Sands^a ^*

^a Department of Chemistry, University of Kentucky, Chemistry–Physics Building, Lexington, Kentucky 40506-0055, USA
Correspondence e-mail: sands@pop.uky.edu

3.1.2. Scalar product

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The scalar product of vectors u and v is defined as $[{\bf u} \cdot {\bf v} = uv \cos \varphi, \eqno(3.1.2.1)]$ where u and v are the lengths of the vectors and $[\varphi]$ is the angle between them. In terms of components, $[\eqalignno{ &{\bf u} \cdot {\bf v} = (u^{i}{\bf a}_{i}) \cdot (v\hskip 2pt^{j}{\bf a}_{j}) &(3.1.2.2)\cr &{\bf u} \cdot {\bf v} = u^{i}v\hskip 2pt^{j}{\bf a}_{i} \cdot {\bf a}_{j} &(3.1.2.3)\cr &{\bf u} \cdot {\bf v} = u^{i}v\hskip 2pt^{j}g_{ij}. &(3.1.2.4)}%(3.1.2.4)]$ In all equations in this chapter, the convention is followed that summation is implied over an index that is repeated once as a subscript and once as a superscript in an expression; thus, the right-hand side of (3.1.2.4) implies the sum of nine terms $[u^{1}v^{1}g_{11} + u^{1}v^{2}g_{12} + \ldots + u^{3}v^{3}g_{33}.]$ The $[g_{ij}]$ in (3.1.2.4) are the components of the metric tensor [see Chapter 1.1 and Sands (1982a)] $[g_{ij} = {\bf a}_{i} \cdot {\bf a}_{j}. \eqno(3.1.2.5)]$ Subscripts are used for quantities that transform the same way as the basis vectors $[{\bf a}_{i}]$ ; such quantities are said to transform covariantly. Superscripts denote quantities that transform the same way as coordinates $[x^{i}]$ ; these quantities are said to transform contravariantly (Sands, 1982a).

Equation (3.1.2.4) is in a form convenient for computer evaluation, with indices i and j taking successively all values from 1 to 3. The matrix form of (3.1.2.4) is useful both for symbolic manipulation and for computation, $[{\bf u} \cdot {\bf v} = {\bi u^{T}} {\bi gv}, \eqno(3.1.2.6)]$ where the superscript italic T following a matrix symbol indicates a transpose. Written out in full, (3.1.2.6) is $[{\bf u} \cdot {\bf v} = (u^{1} u^{2} u^{3}) \pmatrix{g_{11} &g_{12} &g_{13}\cr g_{21} &g_{22} &g_{23}\cr g_{31} &g_{32} &g_{33}\cr} \pmatrix{v^{1}\cr v^{2}\cr v^{3}\cr}. \eqno(3.1.2.7)]$ If u is the column vector with components $[u^{1}, u^{2}, u^{3}]$ , $[ {\bi u}^{T}]$ is the corresponding row vector shown in (3.1.2.7).

References

Sands, D. E. (1982a). Vectors and tensors in crystallography. Reading: Addison Wesley. Reprinted (1995) Dover Publications.Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 3.1, p. 348