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

NOTE 1 In an n-dimensional space with orthonormal base vectors the scalar product of two vectors U and V is the sum of the products of each coordinate $U_i$ of the vector U and the corresponding coordinate $V_i$ of the vector V:

$U\cdot V={\displaystyle \sum _i{U}_i}{V}_i$

NOTE 2 For two complex vectors U and V either the scalar product $U\cdot V$ or a Hermitian product $U\cdot V*$ may be used depending on the application.

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