# N-Dimensional Vector Space

## N-Dimensional Vector Space

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.