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

Language: | en | Status: Standard | |

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 to entry: 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 to entry: 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 to entry: 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 to entry: 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 to entry: The scalar product is denoted by a half-high dot (·) between the two symbols representing the vectors. | ||

Publication date: | 2008-08 | ||

Source: | |||

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Internal notes: | 2017-02-20: Editorial revisions in accordance with the information provided in C00019 (IEV 102) - evaluation. JGO 2017-08-24: replaced a ">" in 2nd bullet point with a ">". LMO | ||

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

Note 1 to entry: 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 to entry: For two complex vectors ** U** and

Note 3 to entry: 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 to entry: 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 to entry: The scalar product is denoted by a half-high dot (·) between the two symbols representing the vectors.