# Scalar Product

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## scalar product

[′skā·lər ′präd·əkt]## 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) (α**a, 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 = (a_{1}, a_{2}, a_{3}) and b = (b_{1}, b_{2}, b_{3}), then (a, b) = a_{1}b_{1} + a_{2} b_{2} + a_{3} b_{3}.

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 (*see*COMPLETE 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 a_{1}**i** + a_{2}* j +* a

_{3}

**k**and b

_{1}

**i**+ b

_{2}

**j**+ b

_{3}

*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.*

**k**.