A geometric vector space in the field [latex]\mathbb{R} [/latex] of real numbers [latex]\mathbb{R}[/latex] is a set of geometric vectors V of the plane or space that has an internal binary operation, called addition, that is an application of V × V in V, and that associates with each pair [latex](\overrightarrow {u}, \overrightarrow {v})[/latex], the vector sum [latex]\overrightarrow {u}\space+\space\overrightarrow {v}[/latex], and an external binary operation, called "multiplication by a scalar", that is an application of [latex]\mathbb{R}×V[/latex] in V, and that associates with each pair [latex](c,\overrightarrow {u})[/latex] the vector product [latex]c\overrightarrow{u}[/latex]. Therefore, an n-dimensional Euclidean space is an n-dimensional vector space with a scalar product.