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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.
Another method of deriving vector and tensor derivative identities is to replace all occurrences of a vector in an algebraic identity by the del operator, provided that no variable occurs both inside and outside the scope of an operator or both inside the scope of one operator in a term and outside the scope of another operator in the same term ...
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.
The product rule extends to various product operations of vector ... The only properties of multiplication used in ... the product rule is the defining property ...
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 ...
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 regressive product, like the exterior product, is associative. [28] The inner product on vectors can also be generalized, but in more than one non-equivalent way. The paper gives a full treatment of several different inner products developed for geometric algebras and their interrelationships, and the notation is taken from there. Many ...
In abstract algebra, the triple product property is an identity satisfied in some groups. Let G {\displaystyle G} be a non-trivial group. Three nonempty subsets S , T , U ⊂ G {\displaystyle S,T,U\subset G} are said to have the triple product property in G {\displaystyle G} if for all elements s , s ′ ∈ S {\displaystyle s,s'\in S} , t , t ...