# Scalar Product

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## 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 ?
A set of 3-phasors composes a 3D complex space C(3) with complex scalar product [1]
The complex power (39) of the vector harmonics of the current and voltage of the k-th order equals to the complex scalar product (27) of the 3-phasor of the voltage and the 3-phasor of the current of the k-th order in the complex 3D space
Multiplying (8) by Q(t) and doing the scalar product, we can write (Q(t)u'(t),u(t)) = (Q(t)A(t)u(t),u(t)).
The common value of all the above scalar products will be denoted by ([omega] | [eta]), without any reference to chiralities.
As explained before, the minima of the scalar products in a neigbourhood are used to detect sudden changes of local orientations, characterizing for the cellulose case the boundaries of crystalline fluxes (cf.
Having fixed a certain direction, we should consider the scalar product of the unit vector of this direction and gradient vector [F'.
In a multidimensional space, the simple product is replaced by the scalar product, that is, the sum of the dimensionwise products.
N] it appears to be natural to use the aggregated scalar product associated with a scalar product [<*, *>.
Not to overburden the notation, since the scalar products of this section will all be in the space H, we are allowed to drop the relative subscript.

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