Set whose elements are vectors in an n-dimensional space.
A geometric vector space in the field \(\mathbb{R} \) of real numbers \(\mathbb{R}\) 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 \((\overrightarrow {u}, \overrightarrow {v})\), the vector sum \(\overrightarrow {u}\space+\space\overrightarrow {v}\), and an external binary operation, called “multiplication by a scalar”, that is an application of \(\mathbb{R}×V\) in V, and that associates with each pair \((c,\overrightarrow {u})\) the vector product \(c\overrightarrow{u}\).
Therefore, an n-dimensional Euclidean space is an n-dimensional vector space with a scalar product.
Examples
- The plane in classical geometry is a two-dimensional Euclidean vector space.
- Physical space is a three-dimensional Euclidean vector space.