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scalar triple product

Triple product

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.

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.

Geometric interpretation

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})

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.

Scalar or pseudoscalar

See also: Cross product and handedness
The scalar triple product typically returns a pseudoscalar, although a pseudoscalar is equivalent to a (true) scalar if the (mathematical) orientation of the coordinate system is selected in advance and fixed.

More exactly, a · (b × c) is a (true) scalar only if:

  • both a and b × c are (true) vectors, or
  • they are both pseudovectors.

Otherwise, it is a pseudoscalar. For instance, if a, b, and c are all vectors, then b × c yields a pseudovector, and a · (b × c) returns a pseudoscalar.

Scalar triple product as an exterior product

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 abc as the parallelepiped spanned by a, b, and c, with the bivectors ab, ac and bc 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 abc (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}cdotmathbf{c}) - mathbf{c}(mathbf{a}cdotmathbf{b})
(mathbf{a}times mathbf{b})times mathbf{c} = -mathbf{c}times(mathbf{a}times mathbf{b}) = - (mathbf{b}cdotmathbf{c})mathbf{a} + (mathbf{a}cdotmathbf{c}) mathbf{b}.

The first formula is known as triple product expansion, or Lagrange's formula. Its right hand member is easier to remember by using the mnemonic “BAC minus CAB”, provided you keep 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 Delta = d delta + delta 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_{kell m} b^ell c^m = varepsilon_{ijk}varepsilon_{kell m} a^j b^ell c^m

Note

See also

References

  • Lass, Harry Vector and Tensor Analysis. McGraw-Hill Book Company, Inc..

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