Tables for
Volume D
Physical properties of crystals
Edited by A. Authier

International Tables for Crystallography (2013). Vol. D, ch. 1.1, pp. 9-10

Section Outer product

A. Authiera*

aInstitut de Minéralogie et de Physique des Milieux Condensés, 4 Place Jussieu, 75005 Paris, France
Correspondence e-mail: Outer product

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The tensor defined by [{\bf x} \bigwedge {\bf y} = {\bf x} \otimes {\bf y} - {\bf y} \otimes {\bf x}]is called the outer product of vectors x and y. (Note: The symbol is different from the symbol [\wedge] for the vector product.) The analytical expression of this tensor of rank 2 is [\left. \matrix{{\bf x} & =x^{i}{\bf e}_{i} \cr {\bf y} & =y\hskip1pt^{j}{\bf e}_{j} \cr}\right\} \quad \Longrightarrow \quad{\bf x}\bigwedge {\bf y} = (x^{i}y\hskip1pt^{j}- y^{i}x\hskip1pt^{j})\, {\bf e}_{i} \otimes {\bf e}_{j}.]

The components [p^{ij} = x^{i}y\hskip1pt^{j}- y^{i}x\hskip1pt^{j}] of this tensor satisfy the properties [p^{ij}= - p\hskip1pt^{ji} ;\quad p^{ii}= 0.]It is an antisymmetric tensor of rank 2. Vector product

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Consider the so-called permutation tensor of rank 3 (it is actually an axial tensor – see Section[link]) defined by [\left\{ \matrix{\varepsilon _{ijk} = + 1 &\hbox{ if the permutation } ijk \hbox{ is even}\hfill\cr \varepsilon _{ijk} = - 1 &\hbox{ if the permutation } ijk \hbox{ is odd}\hfill\cr \varepsilon _{ijk} = 0\hfill &\hbox{ if at least two of the three indices are equal}\cr} \right.]and let us form the contracted product [z_{k}= \textstyle{1\over 2} \varepsilon_{ijk}p^{ij}=\varepsilon_{ijk}x^iy^j. \eqno(]It is easy to check that[\left\{\matrix{z_{1} = x^{2} y^{3} - y^{2} x^{3}\cr z_{2} = x^{3} y^{1} - y^{3} x^{1}\cr z_{3} = x^{1} y^{2} - y^{2} x^{1}.\cr}\right.]

One recognizes the coordinates of the vector product. Properties of the vector product

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Expression ([link] of the vector product shows that it is of a covariant nature. This is indeed correct, and it is well known that the vector product of two vectors of the direct lattice is a vector of the reciprocal lattice [see Section 1.1.4[link] of Volume B of International Tables for Crystallography (2008)[link]].

The vector product is a very particular vector which it is better not to call a vector: sometimes it is called a pseudovector or an axial vector in contrast to normal vectors or polar vectors. The components of the vector product are the independent components of the antisymmetric tensor [p_{ij}]. In the space of n dimensions, one would write [v_{i_{3}i_{4}\ldots i_{n}} = \textstyle{1 \over 2} \varepsilon_{i_{1}i_{2}\ldots i_{n}} p^{i_{1}i_{2}}.]

The number of independent components of [p^{ij}] is equal to [(n^{2} - n)/2] or 3 in the space of three dimensions and 6 in the space of four dimensions, and the independent components of [p^{ij}] are not the components of a vector in the space of four dimensions.

Let us also consider the behaviour of the vector product under the change of axes represented by the matrix [\pmatrix{\bar{1} & 0 & 0\cr 0 &\bar{1} &0\cr 0 &0 &\bar{1}\cr}.]

This is a symmetry with respect to a point that transforms a right-handed set of axes into a left-handed set and reciprocally. In such a change, the components of a normal vector change sign. Those of the vector product, on the contrary, remain unchanged, indicating – as one well knows – that the orientation of the vector product has changed and that it is not, therefore, a vector in the normal sense, i.e. independent of the system of axes.


International Tables for Crystallography (2008). Vol. B, Reciprocal Space, edited by U. Shmueli. Heidelberg: Springer.

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