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IEVref: | 102-03-36 | ID: | |

Language: | en | Status: Standard | |

Term: | vector product | ||

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Definition: | axial vector $U\times V$, orthogonal to two given vectors and U, such that the three vectors V, U and $U\times V$ form a right-handed trihedron or a left-handed trihedron according to the space orientation, with its magnitude equal to the product of the magnitudes of the given vectors and the sine of the angle $\vartheta $ between themV$\left|U\times V\right|=\left|U\right|\cdot \left|V\right|\cdot \mathrm{sin}\vartheta $ Note 1 to entry: In the three-dimensional Euclidean space with given space orientation, the vector product of two vectors is the unique axial vector $U\times V$ such that for any vector V in the same vector space the scalar triple product (W,U,V) is equal to the scalar product $(U\times V)\cdot W$. WNote 2 to entry: For two vectors $U={U}_{x}{e}_{x}+{U}_{y}{e}_{y}+{U}_{z}{e}_{z}$ and $V={V}_{x}{e}_{x}+{V}_{y}{e}_{y}+{V}_{z}{e}_{z}$, where ${e}_{x}\text{,}{e}_{y}\text{,}{e}_{z}$ is an orthonormal base, the vector product is expressed by $U\times V=({U}_{y}{V}_{z}-{U}_{z}{V}_{y}){e}_{x}+({U}_{z}{V}_{x}-{U}_{x}{V}_{z}){e}_{y}+({U}_{x}{V}_{y}-{U}_{y}{V}_{x}){e}_{z}$. The vector product can also be expressed as $U\times \text{}V=\left|\begin{array}{ccc}{e}_{x}& {e}_{y}& {e}_{z}\\ {U}_{x}& {U}_{y}& {U}_{z}\\ {V}_{x}& {V}_{y}& {V}_{z}\end{array}\right|$ using a sum similar to the sum used to obtain the determinant of a matrix. The vector product is therefore the axial vector associated with the antisymmetric tensor $U\otimes V-V\otimes U$ (see IEV 102-03-43). Note 3 to entry: For two complex vectors , either the vector product $U\times V$ or one of the vector products ${U}^{*}\times V$ or $U\times {V}^{*}$ may be used depending on the application. VNote 4 to entry: A vector product can be similarly defined for a pair consisting of a polar vector and an axial vector and is then a polar vector, or for a pair of two axial vectors and is then an axial vector. Note 5 to entry: In the usual three-dimensional space, the vector product of two vector quantities is the vector product of the associated unit vectors multiplied by the product of the scalar quantities. Note 6 to entry: The vector product operation is denoted by a cross (×) between the two symbols representing the vectors. The use of the symbol ∧ is deprecated. | ||

Publication date: | 2008-08 | ||

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Internal notes: | 2017-02-20: Editorial revisions in accordance with the information provided in C00019 (IEV 102) - evaluation. JGO | ||

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$\left|U\times V\right|=\left|U\right|\cdot \left|V\right|\cdot \mathrm{sin}\vartheta $

Note 1 to entry: In the three-dimensional Euclidean space with given space orientation, the vector product of two vectors ** U** and

Note 2 to entry: For two vectors $U={U}_{x}{e}_{x}+{U}_{y}{e}_{y}+{U}_{z}{e}_{z}$ and $V={V}_{x}{e}_{x}+{V}_{y}{e}_{y}+{V}_{z}{e}_{z}$, where ${e}_{x}\text{,}{e}_{y}\text{,}{e}_{z}$ is an orthonormal base, the vector product is expressed by $U\times V=({U}_{y}{V}_{z}-{U}_{z}{V}_{y}){e}_{x}+({U}_{z}{V}_{x}-{U}_{x}{V}_{z}){e}_{y}+({U}_{x}{V}_{y}-{U}_{y}{V}_{x}){e}_{z}$.

The vector product can also be expressed as $U\times \text{}V=\left|\begin{array}{ccc}{e}_{x}& {e}_{y}& {e}_{z}\\ {U}_{x}& {U}_{y}& {U}_{z}\\ {V}_{x}& {V}_{y}& {V}_{z}\end{array}\right|$ using a sum similar to the sum used to obtain the determinant of a matrix. The vector product is therefore the axial vector associated with the antisymmetric tensor $U\otimes V-V\otimes U$ (see IEV 102-03-43).

Note 3 to entry: For two complex vectors ** U** and

Note 4 to entry: A vector product can be similarly defined for a pair consisting of a polar vector and an axial vector and is then a polar vector, or for a pair of two axial vectors and is then an axial vector.

Note 5 to entry: In the usual three-dimensional space, the vector product of two vector quantities is the vector product of the associated unit vectors multiplied by the product of the scalar quantities.

Note 6 to entry: The vector product operation is denoted by a cross (×) between the two symbols representing the vectors. The use of the symbol ∧ is deprecated.