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

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

Section 3.1.2. Scalar product

D. E. Sandsa*

aDepartment 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)[link] 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)[link] are the components of the metric tensor [see Chapter 1.1[link] and Sands (1982a[link])] [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[link]).

Equation (3.1.2.4)[link] 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)[link] 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)[link] 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)[link].

References

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








































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