Tables for
Volume B
Reciprocal space
Edited by U. Shumeli

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

Section 3.1.4. Angle between two vectors

D. E. Sandsa*

aDepartment of Chemistry, University of Kentucky, Chemistry–Physics Building, Lexington, Kentucky 40506-0055, USA
Correspondence e-mail:

3.1.4. Angle between two vectors

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By ([link] and ([link], the angle [\varphi] between vectors u and v is given by [\varphi = \cos^{-1} [u^{i} v\hskip 2pt^{j} g_{ij}/(uv)]. \eqno(] An even more concise expression of equations such as ([link] is possible by making use of the ability of the metric tensor g to convert components from contravariant to covariant (Sands, 1982a[link]). Thus, [v_{i} = g_{ij} v\hskip 2pt^{j},\quad u_{j} = g_{ij} u^{i}, \eqno(] and ([link] may be written succinctly as [{\bf u} \cdot {\bf v} = u^{i} v_{i} \eqno(] or [{\bf u} \cdot {\bf v} = u_{i} v^{i}. \eqno(] With this notation, the angle calculation of ([link] becomes [\varphi = \cos^{-1} [u^{i} v_{i}/(uv)] = \cos^{-1} [u_{i} v^{i}/(uv)]. \eqno(] The summations in ([link], ([link] and ([link] include only three terms, and are thus equivalent in numerical effort to the computation in a Cartesian system, in which the metric tensor is represented by the unit matrix and there is no numerical distinction between covariant components and contravariant components.

Appreciation of the elegance of tensor formulations may be enhanced by noting that corresponding to the metric tensor g with components [g_{ij}] there is a contravariant metric tensor [{\bf g}^{*}] with components [g^{ij} = {\bf a}^{i} \cdot {\bf a}\hskip 2pt^{j}. \eqno(] The [{\bf a}^{i}] are contravariant basis vectors, known to crystallographers as reciprocal axes. Expressions parallel to ([link] may be written, in which [{\bf g}^{*}] plays the role of converting covariant components to contravariant components. These tensors thus express mathematically the crystallographic notions of crystal space and reciprocal space [see Chapter 1.1[link] and Sands (1982a[link])].


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

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