This article is about mathematics. See Lawson criterion for the use of the term triple product in relation to nuclear fusion. In Nuclear fusion research the Lawson criterion, first derived by John D In Physics and Nuclear chemistry, nuclear fusion is the process by which multiple- like charged atomic nuclei join together to form a heavier nucleus

In vector calculus, there are two ways of multiplying three vectors together, to make a triple product of vectors. Vector calculus (also called vector analysis) is a field of Mathematics concerned with multivariable Real analysis of vectors in an Inner

Scalar triple product

Three vectors defining a parallelepiped

The scalar triple product is defined as the dot product of one of the vectors with the cross product of the other two. In Mathematics, the dot product, also known as the scalar product, is an operation which takes two vectors over the Real numbers R In Mathematics, the cross product is a Binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which

Geometric interpretation

Main article: Parallelepiped

Geometrically, the scalar triple product

is the (signed) volume of the parallelepiped defined by the three vectors given. Properties Any of the three pairs of parallel faces can be viewed as the base planes of the prism Properties Any of the three pairs of parallel faces can be viewed as the base planes of the prism

Properties

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

The parentheses may be omitted without causing ambiguity, since the dot product cannot be evaluated first. In Mathematics, the dot product, also known as the scalar product, is an operation which takes two vectors over the Real numbers R 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. In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n

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. In Geometry, a set of points in space is coplanar if the points all lie in the same geometric plane.

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. In Mathematics, the cross product is a Binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which In Physics, a pseudoscalar is a quantity that behaves like a scalar, except that it changes sign under a parity inversion such as Improper rotations See also Orientation (geometry. In Mathematics, an orientation on a real Vector space is a choice of which

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

• both a and b × c are (true) vectors, or
• they are both pseudovectors. In Physics and Mathematics, a pseudovector (or axial vector) is a quantity that transforms like a vector under a proper rotation but gains an

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

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. In Differential geometry, the exterior derivative extends the concept of the differential of a function which is a form of degree zero to Differential forms In Differential geometry, a p -vector is the Tensor obtained by taking Linear combinations of the Wedge product of p In Differential geometry, a p -vector is the Tensor obtained by taking Linear combinations of the Wedge product of p 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). In Mathematics, the Hodge star operator or Hodge dual is a significant Linear map introduced in general by W

Vector triple product

The vector triple product is defined as the cross product of one vector with the cross product of the other two. In Mathematics, the cross product is a Binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which The following relationships hold:

The first formula is known as triple product expansion, or Lagrange's formula^[1]. Lagrange's formula may refer to a number of results named after Joseph Louis Lagrange: Lagrange's interpolation formula - Lagrange polynomial 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. A mnemonic device (nəˈmɒnɪk is a Memory aid Commonly met mnemonics are often verbal something such as a very short poem or a special word used to help a person remember

These formulas are very useful in simplifying vector calculations in physics. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. A related identity regarding gradients and useful in vector calculus is

This can be also regarded as a special case of the more general Laplace-de Rham operator Δ = dδ + δd. In Vector calculus, the gradient of a Scalar field is a Vector field which points in the direction of the greatest rate of increase of the scalar Vector calculus (also called vector analysis) is a field of Mathematics concerned with multivariable Real analysis of vectors in an Inner In Mathematics and Physics, the Laplace operator or Laplacian, denoted by \Delta\  or \nabla^2  and named after

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. In Mathematics, the cross product is a Binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which In Physics and Mathematics, a pseudovector (or axial vector) is a quantity that transforms like a vector under a proper rotation but gains an 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.

Note

1. ^ 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). In Algebra, Lagrange's identity is the identity \biggl( \sum_{k=1}^n a_k^2\biggr \biggl(\sum_{k=1}^n b_k^2\biggr - \biggl(\sum_{k=1}^n a_k b_k\biggr^2 Encyclopedic Dictionary of Mathematics. MIT Press, p. 1679. ISBN 0262590204.  

See also

• Jacobi triple product

References

• Lass, Harry (1950). In Mathematics, the Jacobi triple product is a relation that re-expresses the Jacobi Theta function, normally written as a series as a product Vector and Tensor Analysis. McGraw-Hill Book Company, Inc. , pp. 23-25.