| axial vector , orthogonal to two given vectors U and V, such that the three vectors U, V and 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 between them|
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 such that for any vector W in the same vector space the scalar triple product (U,V,W) is equal to the scalar product .
Note 2 to entry: For two vectors and , where is an orthonormal base, the vector product is expressed by .
The vector product can also be expressed as 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 (see IEV 102-03-43).
Note 3 to entry: For two complex vectors U and V, either the vector product or one of the vector products or 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.