(Untitled) | (Untitled) | (Untitled) | (Untitled) | (Untitled) | Examples |

IEVref: | 102-03-17 | ID: | |

Language: | en | Status: backup | |

Term: | scalar product | ||

Synonym1: | dot product [Preferred] | ||

Synonym2: | |||

Synonym3: | |||

Symbol: | |||

Definition: | scalar, denoted by $U\cdot V$, attributed to any pair of vectors and U in a vector space by a given bilinear form, with the following properties: V- symmetry: $U\cdot V=V\cdot U$,
- $U\cdot U>0$ for $U\ne 0$
NOTE 1 In an is the sum of the products of each coordinate ${U}_{i}$ of the vector V and the corresponding coordinate ${V}_{i}$ of the vector U: V$U\cdot V={\displaystyle \sum _{i}{U}_{i}}{V}_{i}$ NOTE 2 For two complex vectors either the scalar product $U\cdot V$ or a Hermitian product $U\cdot V*$ may be used depending on the application. VNOTE 3 A scalar product can be similarly defined for a pair consisting of a polar vector and an axial vector and is then a pseudo-scalar, or for a pair of two axial vectors and is then a scalar. NOTE 4 The scalar product of two vector quantities is the scalar product of the associated unit vectors multiplied by the product of the scalar quantities. NOTE 5 The scalar product is denoted by a half-high dot (·) between the two symbols representing the vectors. | ||

Publication date: | 2007-08 | ||

Source: | |||

Replaces: | |||

Internal notes: | |||

CO remarks: | |||

TC/SC remarks: | |||

VT remarks: | |||

Domain1: | |||

Domain2: | |||

Domain3: | |||

Domain4: | |||

Domain5: |

- symmetry: $U\cdot V=V\cdot U$,
- $U\cdot U>0$ for $U\ne 0$

NOTE 1 In an *n*-dimensional space with orthonormal base vectors the scalar product of two vectors ** U** and

$U\cdot V={\displaystyle \sum _{i}{U}_{i}}{V}_{i}$

NOTE 2 For two complex vectors ** U** and

NOTE 3 A scalar product can be similarly defined for a pair consisting of a polar vector and an axial vector and is then a pseudo-scalar, or for a pair of two axial vectors and is then a scalar.

NOTE 4 The scalar product of two vector quantities is the scalar product of the associated unit vectors multiplied by the product of the scalar quantities.

NOTE 5 The scalar product is denoted by a half-high dot (·) between the two symbols representing the vectors.

102-03-17en.gif |