Triple product
From Wikipedia, the free encyclopedia
- This article is about mathematics. See Lawson criterion for the use of the term triple product in relation to nuclear fusion.
In vector calculus, there are two ways of multiplying three vectors together, to make a triple product of vectors.
Contents |
[edit] Scalar triple product
The scalar triple product is defined as the dot product of one of the vectors with the cross product of the other two. It is a scalar (more precisely, it can be either a scalar or a pseudoscalar).
Geometrically, this product is the (signed) volume of the parallelepiped formed by the three vectors given. It can be evaluated numerically using any one of the following equivalent characterizations:
- <math>
\mathbf{a}\cdot(\mathbf{b}\times \mathbf{c})= \mathbf{b}\cdot(\mathbf{c}\times \mathbf{a})= \mathbf{c}\cdot(\mathbf{a}\times \mathbf{b}) </math>
The parentheses may be omitted without causing ambiguity, since the dot product cannot be evaluated first. If it were, it would leave the cross product of a vector and a scalar, which is not defined.
The scalar triple product can also be understood as the determinant of the 3-by-3 matrix having the three vectors as rows (or columns, since the determinant for a transposed matrix, is the same as the original); this quantity is invariant under coordinate rotation.
Another useful property of the scalar triple product is that if it is equal to zero, then the three vectors a, b, and c are coplanar.
More generally, whether the scalar triple product is a (true) scalar or a pseudoscalar is defined by matching the possible vector combinations in the cross product according to the rules given in cross product and handedness. If and only if the result of the cross product is a (true) vector, the scalar triple product is a (true) scalar.
[edit] Scalar triple product as an exterior product
The triple product can be viewed in terms of the exterior product in a similar way to the cross product. In exterior calculus the exterior product of two vectors is a bivector, while the exterior product of three vectors is a trivector. A bivector is an oriented plane element, while a trivector is an oriented volume element, in much the same way that a vector is an oriented line element. Given vectors a, b and c, one can view the trivector a∧b∧c as the parallelepiped spanned by a, b, and c, with the bivectors a∧b, a∧c and b∧c forming three of the 6 faces of the parallelepiped. We obtain the triple product by taking the Hodge dual of the trivector a∧b∧c (in much the same way that the cross product can be obtained by taking the Hodge dual of a bivector). The Hodge dual can be thought of as the oriented multi-dimensional element "perpendicular" to the trivector. In three dimensions (and three dimensions only) this results in a scalar value (there being no dimensions left to be "perpendicular" to the volume element). The magnitude of the scalar is given by the magnitude of the trivector; that is, the size of the trivector a∧b∧c relative to the unit trivector (i.e. the unit volume), which gives the triple product as the volume of the parallelepiped as expected.
[edit] Vector triple product
- See also: Lagrange's formula
The vector triple product is defined as the cross product of one vector with the cross product of the other two.
The following relationships hold:
- <math>\mathbf{a}\times (\mathbf{b}\times \mathbf{c}) = \mathbf{b}(\mathbf{a}\cdot\mathbf{c}) - \mathbf{c}(\mathbf{a}\cdot\mathbf{b})</math>
- <math>(\mathbf{a}\times \mathbf{b})\times \mathbf{c} = -\mathbf{c}\times(\mathbf{a}\times \mathbf{b}) = - \mathbf{a}(\mathbf{b}\cdot\mathbf{c}) + \mathbf{b}(\mathbf{a}\cdot\mathbf{c})</math>
The vector triple product may be a vector or a pseudovector. The case is defined by matching the possible vector combinations in each of the two cross products according to the rules given in cross product. For example, if all three are vectors, the result is a vector. But if one or all of the three are the pseudovectors, the result is a pseudovector.
[edit] See also
[edit] References
- Lass, Harry (1950). Vector and Tensor Analysis. McGraw-Hill Book Company, Inc., pp. 23-25. cs:Smíšený součin
de:Spatprodukt fr:Calcul vectoriel en géométrie euclidienne#Produit_mixte he:מכפלה מעורבת pl:Iloczyn mieszany wektorów pt:Produto triplo ru:Смешанное произведение zh:三重积
