Triple product: Information from Answers.com

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.

1 Scalar triple product
2 Vector triple product
- 2.1 Vector or pseudovector
3 Notation
4 Note
5 See also
6 References

Scalar triple product

Three vectors defining a parallelepiped

The scalar triple product (also called mixed product) is defined as the dot product of one of the vectors with the cross product of the other two.

Geometric interpretation

Main article: Parallelepiped

Geometrically, the scalar triple product

$\mathbf{a}\cdot(\mathbf{b}\times \mathbf{c})$

is the (signed) volume of the parallelepiped defined by the three vectors given.

Properties

The scalar triple product can be evaluated numerically using any one of the following equivalent characterizations:

$\mathbf{a}\cdot(\mathbf{b}\times \mathbf{c})= \mathbf{b}\cdot(\mathbf{c}\times \mathbf{a})= \mathbf{c}\cdot(\mathbf{a}\times \mathbf{b})$

Switching the two vectors in the cross product negates the triple product, i.e.:

$\mathbf{c}\cdot(\mathbf{a}\times \mathbf{b})=-\mathbf{c}\cdot(\mathbf{b}\times \mathbf{a})$ .

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 3x3 matrix having the three vectors as its rows or columns (the determinant of a transposed matrix is the same as the original); this quantity is invariant under coordinate rotation.

$\mathbf{a}\cdot(\mathbf{b}\times \mathbf{c}) = \det \begin{bmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \\ \end{bmatrix}.$

Note that if the scalar triple product is equal to zero, the three vectors a, b, and c are coplanar, since the "parallelepiped" defined by them would be flat and have no volume.

There is also this property of triple products:

$[\mathbf{a}\cdot(\mathbf{b}\times \mathbf{c})] \mathbf{a}=(\mathbf{a}\times \mathbf{b})\times (\mathbf{a}\times \mathbf{c})$

Scalar or pseudoscalar

Main article: Pseudoscalar

Scalar triple product as an exterior product

A trivector is an oriented volume element; its Hodge dual is a scalar with magnitude equal to its volume.

The scalar triple product can be viewed in terms of the exterior 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. 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.

Given vectors a, b and c, the triple product is the Hodge dual of the trivector a∧b∧c (in much the same way that the cross product is the Hodge dual of a bivector).

Vector triple product

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:

$\mathbf{a}\times (\mathbf{b}\times \mathbf{c}) = \mathbf{b}(\mathbf{a}\cdot\mathbf{c}) - \mathbf{c}(\mathbf{a}\cdot\mathbf{b})$

$(\mathbf{a}\times \mathbf{b})\times \mathbf{c} = -\mathbf{c}\times(\mathbf{a}\times \mathbf{b}) = - (\mathbf{b}\cdot\mathbf{c})\mathbf{a} + (\mathbf{a}\cdot\mathbf{c}) \mathbf{b}.$

The first formula is known as triple product expansion, or Lagrange's formula.^[1] Its right hand member is easier to remember by using the mnemonic “BAC minus CAB”, provided one keeps in mind which vectors are dotted together.

These formulas are very useful in simplifying vector calculations in physics. A related identity regarding gradients and useful in vector calculus is

$\begin{align} \nabla \times (\nabla \times \mathbf{f}) & {}= \nabla (\nabla \cdot \mathbf{f} ) - (\nabla \cdot \nabla) \mathbf{f} \\ & {}= \mbox{grad }(\mbox{div } \mathbf{f} ) - \mbox{laplacian } \mathbf{f}. \end{align}$

This can be also regarded as a special case of the more general Laplace-de Rham operator $Δ = d δ + δ d$ .

Vector or pseudovector

A vector triple product typically returns a (true) vector. More exactly, according to the rules given in cross product and handedness, the triple product a × (b × c) is a vector if either a or b × c (but not both) are pseudovectors. Otherwise, it is a pseudovector. For instance, if a, b, and c are all vectors, then b × c yields a pseudovector, and a × (b × c) returns a vector.

Notation

Using the Levi-Civita symbol, the triple product is

$(\mathbf{a} \cdot (\mathbf{b}\times \mathbf{c})) = \varepsilon_{ijk} a^i b^j c^k$

and

$(\mathbf{a} \times (\mathbf{b}\times \mathbf{c}))_i = \varepsilon_{ijk} a^j \varepsilon_{k\ell m} b^\ell c^m = \varepsilon_{ijk}\varepsilon_{k\ell m} a^j b^\ell c^m$

Note

^ Joseph Louis Lagrange did not develop the cross product as an algebraic product on vectors, but did use an equivalent form of it in components: see Lagrange, J-L (1773). "Solutions analytiques de quelques problèmes sur les pyramides triangulaires". Oeuvres. vol 3. He may have written a formula similar to the triple product expansion in component form. See also Lagrange's identity and Kiyoshi Ito (1987). Encyclopedic Dictionary of Mathematics. MIT Press. p. 1679. ISBN 0262590204.

References

Lass, Harry (1950). Vector and Tensor Analysis. McGraw-Hill Book Company, Inc.. pp. 23–25.

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Triple product

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