# Scalar Product

(redirected from scalar products)
Also found in: Dictionary, Thesaurus.

## scalar product

[′skā·lər ′präd·əkt]
(mathematics)
A symmetric, alternating, or Hermitian form.

## Scalar Product

(or inner product). The scalar product of the two vectors a and b is the scalar that is the product of the lengths of the vectors and the cosine of the angle between the vectors. This product is denoted by (a, b) or a·b and, because of the second notation, is often called the dot product. An example of a scalar product is the work done by a constant force F along a straight path S: this work is equal to (F, S).

The fundamental properties of the scalar product are as follows: (1) (a, b) = (b, a); (2) (αa, b) = α (a, b), where α is a scalar; (3) (a, b + c) = (a, b) + (a, c); (4) (a, a) > 0 if a ≠ 0, and (a, a) = 0 if a = 0. The length of the vector a is equal to , If (a, b) = 0, then a = 0, b = 0, or ab. If, in rectangular Cartesian coordinates, a = (a1, a2, a3) and b = (b1, b2, b3), then (a, b) = a1b1 + a2 b2 + a3 b3.

The concept of the scalar product can be extended to n-dimensional vector spaces. Here, the scalar product is defined by the equality Such geometric concepts as the length of a vector and the angle between two vectors are introduced on the basis of this definition of the scalar product. An infinite-dimensional linear space in which the scalar product is defined and the axiom of completeness is satisfied with respect to the norm (seeCOMPLETE METRIC SPACE) is called a Hilbert space. Hilbert spaces play an important role in functional analysis and quantum mechanics. For vector spaces over the field of complex numbers, condition (1) is replaced by the condition (a, b) = (b, a), and the scalar product is defined as Two three-dimensional vectors a and b can be regarded as the pure quaternions a1i + a2j + a3k and b1i + b2j + b3k. The scalar product of a and b is equal to the negative of the scalar part of the product of the quaternions. The vector product of a and b, it may be noted, is equal to the vector part of the product of the quaternions.

References in periodicals archive ?
For that we need to endow linear spaces with a scalar product. This additional structure greatly enlarges the domain of applications of linear algebra.
Due to the compactness of a sphere in a finite-dimensional space and the completeness and the symmetric property of the dictionary it implies that scalar products
In Euclidean space the scalar product of two constant fields does not depend on the point where it was computed.
(1) The client obtains KnnRes.MaxDist from the list KnnRes and verifies that any scalar product in the KnnRes is less than or equal to KnnRes.MaxDist, while the other scalar products are greater than KnnRes.MaxDist.
Representative SMCs include secure comparison, secure sum, and secure scalar product, and SSDD is an interesting application of exploiting the secure scalar product.
For 3-phase vector curves x(t), y(t) [member of] [L.sup.(3).sub.2] (T) a scalar product is determined
As a result, we can replace the scalar products of the PCTC by their initial values in the case of inter-channel interactions as well.
The significance of the Riesz map or the scalar product, respectively, is pointed out in many publications to varying degrees.
Recall that the scalar product of t and [mu] is a fuzzy set t[mu] [member of] F(U) defined by t[mu](x) = t [conjunction] [mu](x) for all x [member of] U.
Pythagoras' theorem in diagrams like Figure 2 and matrix scalar products  allowed one to associate scalar strengths with residual powers as a tolerance.
<!-- Inner Product --> <CDDefinition> <Name>inner_product</Name> <Role>application</Role> <Description> Inner product specific to <a xmlns="http://www.w3.org/1999/xhtml" href="http://en.wikipedia.org/wiki/Geometric_algebra"> Geometric Algebra </a> is the generalization of the scalar product, defined in CD linalg1, for arbitrary multivectors.
The lack of references to chiralities in the quaternionic scalar products is intentional.

Site: Follow: Share:
Open / Close