IEVref: 102-03-36 ID: Language: en Status: Standard Term: vector product Synonym1: Synonym2: Synonym3: Symbol: Definition: axial vector $U×V$, orthogonal to two given vectors U and V, such that the three vectors U, V and $U×V$ 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 them$|U×V|=|U|\cdot |V|\cdot \mathrm{sin}\vartheta$Note 1 to entry: In the three-dimensional Euclidean space with given space orientation, the vector product of two vectors U and V is the unique axial vector $U×V$ such that for any vector W in the same vector space the scalar triple product (U,V,W) is equal to the scalar product $\left(U×V\right)\cdot W$. 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 $U×V=\left({U}_{y}{V}_{z}-{U}_{z}{V}_{y}\right){e}_{x}+\left({U}_{z}{V}_{x}-{U}_{x}{V}_{z}\right){e}_{y}+\left({U}_{x}{V}_{y}-{U}_{y}{V}_{x}\right){e}_{z}$. The vector product can also be expressed as $U×\text{}V=|\begin{array}{ccc}{e}_{x}& {e}_{y}& {e}_{z}\\ {U}_{x}& {U}_{y}& {U}_{z}\\ {V}_{x}& {V}_{y}& {V}_{z}\end{array}|$ 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 V, either the vector product $U×V$ or one of the vector products ${U}^{*}×V$ or $U×{V}^{*}$ may be used depending on the application. 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. Publication date: 2008-08 Source: Replaces: Internal notes: 2017-02-20: Editorial revisions in accordance with the information provided in C00019 (IEV 102) - evaluation. JGO CO remarks: TC/SC remarks: VT remarks: Domain1: Domain2: Domain3: Domain4: Domain5: