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- 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.

The scalar triple product is defined as the dot product of one of the vectors with the cross product of the other two.

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

- $$

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.

- See also: Cross product and handedness

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.

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).

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

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

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

- $begin\{align\}$

- $(mathbf\{a\}\; cdot\; (mathbf\{b\}times\; mathbf\{c\}))\; =\; varepsilon\_\{ijk\}\; a^i\; b^j\; c^k$

- $(mathbf\{a\}\; times\; (mathbf\{b\}times\; mathbf\{c\}))\_i\; =\; varepsilon\_\{ijk\}\; a^j\; varepsilon\_\{kell\; m\}\; b^ell\; c^m\; =\; varepsilon\_\{ijk\}varepsilon\_\{kell\; m\}\; a^j\; b^ell\; c^m$

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

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Last updated on Sunday September 21, 2008 at 06:52:05 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Sunday September 21, 2008 at 06:52:05 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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