of finite dimension over the reals, or, typically, an element of equipped with the dot product. This makes sense, as the addition in such a vector space acts freely and transitively on the vector space itself. That is, is a Euclidean space, with itself as an associated vector space, and the dot product as an inner product. The Euclidean space is often presented as the Euclidean space of dimension . This is motivated by the fact that every Euclidean space of dimension is isomorphic to the Euclidean space More precisely, given such a Euclidean space, one may choose any point as an origin. By Gram–Schmidt process, one may also find an orthonormal basis of the associated vector space (a basis such that the inner product of two basis vectors is 0 if they are different and 1 if they are equal). This defines Cartesian coordinates of any point of the space, as the coordinates on this basis of the vector These choices define an isomorphism of the given Euclidean space onto by mapping any point to the -tuple of its Cartesian coordinates, and every vector to its coordinate vector. Examples in one dimension Since the physicist's concept of force has a direction and a magnitude, it may be seen as a vector. As an example, consider a rightward force F of 15 newtons. If the positive axis is also directed rightward, then F is represented by the vector 15 N, and if positive points leftward, then the vector for F is −15 N. In either case, the magnitude of the vector is 15 N. Likewise, the vector representation of a displacement Δs of 4 meters would be 4 m or −4 m, depending on its direction, and its magnitude would be 4 m regardless. In physics and engineering Vectors are fundamental in the physical sciences. They can be used to represent any quantity that has magnitude, has direction, and which adheres to the rules of vector addition. An example is velocity, the magnitude of which is speed. For instance, the velocity 5 meters per second upward could be represented by the vector (0, 5) (in 2 dimensions with the positive y-axis as 'up'). Another quantity represented by a vector is force, since it has a magnitude and direction and follows the rules of vector addition. Vectors also describe many other physical quantities, such as linear displacement, displacement, linear acceleration, angular