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In mathematics, the dot product or scalar product [note 1] is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number. In Euclidean geometry , the dot product of the Cartesian coordinates of two vectors is widely used.
A pseudoscalar also results from any scalar product between a pseudovector and an ordinary vector. The prototypical example of a pseudoscalar is the scalar triple product, which can be written as the scalar product between one of the vectors in the triple product and the cross product between the two other vectors, where the latter is a ...
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.
Thus, for example, the product of a 1 × n matrix and an n × 1 matrix, which is formally a 1 × 1 matrix, is often said to be a scalar. The real component of a quaternion is also called its scalar part. The term scalar matrix is used to denote a matrix of the form kI where k is a scalar and I is the identity matrix.
The cross product (also called the vector product or outer product) is only meaningful in three or seven dimensions. The cross product differs from the dot product primarily in that the result of the cross product of two vectors is a vector. The cross product, denoted a × b, is a vector perpendicular to both a and b and is defined as
The direct product of a family of R-algebras is the ring-theoretic direct product. This becomes an R-algebra with the obvious scalar multiplication. Free products One can form a free product of R-algebras in a manner similar to the free product of groups. The free product is the coproduct in the category of R-algebras. Tensor products
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The terms on the right are called scalar of the product, and the vector of the product [51] of two right quaternions. Note: "Scalar of the product" corresponds to Euclidean scalar product of two vectors up to the change of sign (multiplication to −1).