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In geometry and algebra, the triple product is a product of three 3-dimensional vectors, usually Euclidean vectors.The name "triple product" is used for two different products, the scalar-valued scalar triple product and, less often, the vector-valued vector triple product.
There are two lists of mathematical identities related to vectors: Vector algebra relations — regarding operations on individual vectors such as dot product, cross product, etc. Vector calculus identities — regarding operations on vector fields such as divergence, gradient, curl, etc.
The dotted vector, in this case B, is differentiated, while the (undotted) A is held constant. The utility of the Feynman subscript notation lies in its use in the derivation of vector and tensor derivative identities, as in the following example which uses the algebraic identity C⋅(A×B) = (C×A)⋅B:
The following are important identities in vector algebra.Identities that only involve the magnitude of a vector ‖ ‖ and the dot product (scalar product) of two vectors A·B, apply to vectors in any dimension, while identities that use the cross product (vector product) A×B only apply in three dimensions, since the cross product is only defined there.
This category is for mathematical identities, i.e. identically true relations holding in some area of algebra (including abstract algebra, or formal power series). Subcategories This category has only the following subcategory.
In modern geometry, Euclidean spaces are often defined by using vector spaces. In this case, the dot product is used for defining lengths (the length of a vector is the square root of the dot product of the vector by itself) and angles (the cosine of the angle between two vectors is the quotient of their dot product by the product of their ...
The Jacobi triple product identity is the Macdonald identity for the affine root system of type A 1, and is the Weyl denominator formula for the corresponding affine Kac–Moody algebra. Properties [ edit ]
Basis Decomposition of a 2-vector. For vectors in R 3, the exterior algebra is closely related to the cross product and triple product.Using the standard basis {e 1, e 2, e 3}, the exterior product of a pair of vectors