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The scalar triple product of three vectors is defined as = = (). Its value is the determinant of the matrix whose columns are the Cartesian coordinates of the three vectors. It is the signed volume of the parallelepiped defined by the three vectors, and is isomorphic to the three-dimensional special case of the exterior product of three vectors.
A scalar in physics and other areas of science is also a scalar in mathematics, as an element of a mathematical field used to define a vector space.For example, the magnitude (or length) of an electric field vector is calculated as the square root of its absolute square (the inner product of the electric field with itself); so, the inner product's result is an element of the mathematical field ...
A scalar is an element of a field which is used to define a vector space.In linear algebra, real numbers or generally elements of a field are called scalars and relate to vectors in an associated vector space through the operation of scalar multiplication (defined in the vector space), in which a vector can be multiplied by a scalar in the defined way to produce another vector.
As a special case, V may be taken to be K itself and scalar multiplication may then be taken to be simply the multiplication in the field. When V is K n, scalar multiplication is equivalent to multiplication of each component with the scalar, and may be defined as such. The same idea applies if K is a commutative ring and V is a module over K.
A scalar beside a vector (either or both of which may be in parentheses) implies scalar multiplication. The two common operators, a dot and a rotated cross, are also acceptable (although the rotated cross is almost never used), but they risk confusion with dot products and cross products, which operate on two vectors.
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 ...
Police name 19-year-old who died during Storm Éowyn Monday 27 January 2025 18:25 , Bryony Gooch Police Scotland have identified Calum Carmichael as the 19-year-old man who died after his car was ...
The scalar triple product (also called the mixed product, box product, or triple scalar product) is defined as the dot product of one of the vectors with the cross product of the other two. Geometric interpretation