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

International Tables for Crystallography (2010). Vol. B, ch. 3.1, pp. 405-406   | 1 | 2 |

Section 3.1.8. Some vector relationships

D. E. Sandsa*

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

3.1.8. Some vector relationships

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The results developed above lead to several useful relationships between vectors; for derivations, see Sands (1982a[link]).

3.1.8.1. Triple vector product

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[\eqalignno{ {\bf u} \wedge ({\bf v} \wedge {\bf w}) &= ({\bf u} \cdot {\bf w}) {\bf v} - ({\bf u} \cdot {\bf v}) {\bf w} &(3.1.8.1)\cr ({\bf u} \wedge {\bf v}) \wedge {\bf w} &= - ({\bf v} \cdot {\bf w}) {\bf u} + ({\bf u} \cdot {\bf w}) {\bf v}. &(3.1.8.2)}%(3.1.8.2)]

3.1.8.2. Scalar product of vector products

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[({\bf u} \wedge {\bf v}) \cdot ({\bf w} \wedge {\bf z}) = ({\bf u} \cdot {\bf w}) ({\bf v} \cdot {\bf z}) - ({\bf u} \cdot {\bf z}) ({\bf v} \cdot {\bf w}). \eqno(3.1.8.3)]A derivation of this result may be found also in Shmueli (1974[link]).

3.1.8.3. Vector product of vector products

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[\eqalignno{ ({\bf u} \wedge {\bf v}) \wedge ({\bf w} \wedge {\bf z}) &= ({\bf u} \cdot {\bf w} \wedge {\bf z}) {\bf v} - ({\bf v} \cdot {\bf w} \wedge {\bf z}) {\bf u} &(3.1.8.4) \cr ({\bf u} \wedge {\bf v}) \wedge ({\bf w} \wedge {\bf z}) &= ({\bf u} \cdot {\bf v} \wedge {\bf z}) {\bf w} - ({\bf u} \cdot {\bf v} \wedge {\bf w}) {\bf z.} &(3.1.8.5)}%(3.1.8.5)]

References

Sands, D. E. (1982a). Vectors and Tensors in Crystallography. Reading: Addison Wesley. Reprinted (1995) Dover Publications.
Shmueli, U. (1974). On the standard deviation of a dihedral angle. Acta Cryst. A30, 848–849.








































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