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 multivariate real analysis of vectors in two or more dimensions. ...

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). In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors over the real numbers R and returns a real-valued scalar quantity. ... For the cross product in algebraic topology, see Künneth theorem. ... A scalar may be: Look up scalar in Wiktionary, the free dictionary. ... In mathematics, a pseudoscalar in a geometric algebra is the highest-grade basis element of the algebra. ...

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: In geometry, a parallelepiped (now usually pronounced , traditionally[1] in accordance with its etymology in Greek παραλληλ-επίπεδον, a body having parallel planes) is a three-dimensional figure like a cube, except that its faces are not squares but parallelograms. ...

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. In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors over the real numbers R and returns a real-valued scalar quantity. ...

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 square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...

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. A set of points is said to be coplanar if and only if they lie on the same geometric plane. ...

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. For the cross product in algebraic topology, see Künneth theorem. ...

Scalar triple product as an exterior product

The exterior product of three vectors gives and oriented volume element, the Hodge dual of which (in thee dimensions) is a scalar with magnitude equal to the volume of the trivector.

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. Image File history File links Size of this preview: 766 × 600 pixelsFull resolution (1068 × 836 pixel, file size: 93 KB, MIME type: image/png) Visualsing the scalar triple product in terms of an exterior product. ... Image File history File links Size of this preview: 766 × 600 pixelsFull resolution (1068 × 836 pixel, file size: 93 KB, MIME type: image/png) Visualsing the scalar triple product in terms of an exterior product. ... In mathematics, the exterior algebra (also known as the Grassmann algebra) of a given vector space V is a certain unital associative algebra which contains V as a subspace. ... For the cross product in algebraic topology, see Künneth theorem. ... This article may be too technical for most readers to understand. ... A bivector is an element of the antisymmetric tensor product of a tangent space with itself. ... A p-vector in differential geometry is the tensor obtained by taking linear combinations of the wedge product of p tangent vectors, for some integer p ≥ 1. ... In mathematics, the Hodge star operator or Hodge dual is a signficant linear map introduced in general by W. V. D. Hodge. ...

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. This is a well-known and useful formula, a × (b × c) = b(a · c) − c(a · b), which is easier to remember as “BAC minus CAB”. This formula is very useful in simplifying vector calculations in physics. ... For the cross product in algebraic topology, see Künneth theorem. ...

The following relationships hold:

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 of the three is a pseudovector, the result is a pseudovector. In physics and mathematics, a pseudovector (or axial vector) is a quantity that transforms like a vector under a proper rotation, but gains an additional sign flip under an improper rotation (a transformation that can be expressed as an inversion followed by a proper rotation). ... For the cross product in algebraic topology, see Künneth theorem. ...

See also

• Jacobi triple product

In mathematics, the Jacobi triple product is a relation that re-expresses the Jacobi theta function, normally written as a series, as a product. ...

References

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