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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 .
A simple example of a smooth fiber bundle is a Cartesian product of two manifolds. Consider the bundle B 1 := (M × N, pr 1) with bundle projection pr 1 : M × N → M : (x, y) → x. Applying the definition in the paragraph above to find the vertical bundle, we consider first a point (m,n) in M × N.
If the Cartesian product rows × columns is taken, the cells of the table contain ordered pairs of the form (row value, column value). [4] One can similarly define the Cartesian product of n sets, also known as an n-fold Cartesian product, which can be represented by an n-dimensional array, where each element is an n-tuple.
Ternary relations may also be referred to as 3-adic, 3-ary, 3-dimensional, or 3-place. Just as a binary relation is formally defined as a set of pairs , i.e. a subset of the Cartesian product A × B of some sets A and B , so a ternary relation is a set of triples, forming a subset of the Cartesian product A × B × C of three sets A , B and C .
The dot product in Cartesian coordinates (Euclidean space with an orthonormal basis set) is simply the sum of the products of components. In orthogonal coordinates, the dot product of two vectors x and y takes this familiar form when the components of the vectors are calculated in the normalized basis:
The lowest-dimensional forms are 6-polytopes being the Cartesian product of three polygons. The smallest can be written as {3} × {3} × {3} in Schläfli symbols if they are regular, and contains 27 vertices. This is the product of three equilateral triangles and is a uniform polytope. The f-vectors can be computed by (3,3,1) 3 = (27,81,108,81 ...
In mathematics, the Kronecker product, sometimes denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix.It is a specialization of the tensor product (which is denoted by the same symbol) from vectors to matrices and gives the matrix of the tensor product linear map with respect to a standard choice of basis.
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.