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

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 $U\times V$ such that for any vector W in the same vector space the scalar triple product (U,V,W) is equal to the scalar product $(U\times V)\cdot W$.

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 $U\times V=(U_y{V}_z-U_z{V}_y)e_x+(U_z{V}_x-U_x{V}_z)e_y+(U_x{V}_y-U_y{V}_x)e_z$.

The vector product can also be expressed as $U\times V=\left|\begin{array}{ccc}{e}_x& e_y& e_z\\ U_x& U_y& U_z\\ V_x& V_y& V_z\end{array}\right|$ 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 V, 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.

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.