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scalar product[′skā·lər ′präd·əkt]
(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 a ⊥ b. 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.