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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 term scalar component refers sometimes to scalar projection, as, in Cartesian coordinates, the components of a vector are the scalar projections in the directions of the coordinate axes.
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.
In mathematics, a geometric algebra (also known as a Clifford algebra) is an algebra that can represent and manipulate geometrical objects such as vectors.Geometric algebra is built out of two fundamental operations, addition and the geometric product.
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).
where s is the parity of the signature of the scalar product on V, that is, the sign of the determinant of the matrix of the scalar product with respect to any basis. For example, if n = 4 and the signature of the scalar product is either (+ − − −) or (− + + +) then s = −1.
A Clifford algebra is a unital associative algebra that contains and is generated by a vector space V over a field K, where V is equipped with a quadratic form Q : V → K.The Clifford algebra Cl(V, Q) is the "freest" unital associative algebra generated by V subject to the condition [c] = , where the product on the left is that of the algebra, and the 1 on the right is the algebra's ...