Scalar Product


Also found in: Dictionary, Thesaurus, Wikipedia.
Related to Scalar Product: Triple scalar product

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 Scalar Product, 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 Scalar Product (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 ?
As a note, the space [L.sup.2]([S.sup.2]) is a Hilbert space, meaning it is a vector space with a scalar product that is complete with respect to the norm induced by the scalar product.
In [10], a comprehensive overview on principal angles defined by the scalar product induced by B can be found.
and for e = [e.sub.n] the scalar product ([r.sub.n-1], e) is positive, for e = [+ or -] [e.sub.n+j] (j [member of] {1, 2, 3, ...}) we have an inequality
Estivill-Castro, "A new efficient privacy-preserving scalar product protocol," in Proceedings of the Sixth Australasian Conference on Data Mining and Analytics, vol.
Let ( G/K, g) be a generalized Wallach space with reductive decomposition g = [??] [direct sum] m, equipped with a G-invariant metric corresponding to a scalar product of the form (8).
In our scheme, scalar product comparison rather than Euclidean distance comparison is adopted to determine where to insert a data point.
In order to compute the average execution time that the GPU and CPU spend for generating the sample arrays, computing the scalar product and the total execution time, we have used the "StopWatchInterface" offered by the CUDA application programming interface (API) for defining, creating and managing the timers.
Then [parallel]B[parallel] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] denotes the norm associated with the scalar product [([alpha], [beta]).sub.[L.sub.2]([GAMMA])].
Xu, "A protocol for the secure two-party quantum scalar product," Physics Letters A, vol.
This square root of the scalar product <[G.sub.1], [G.sub.1]> [8] is independent of the meridians in [Q.sub.p].
<!-- 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.