as defined in pure mathematics, they are sometimes referred to as geometric, spatial, or Euclidean vectors. Being an arrow, a Euclidean vector possesses a definite initial point and terminal point. A vector with fixed initial and terminal point is called a bound vector. When only the magnitude and direction of the vector matter, then the particular initial point is of no importance, and the vector is called a free vector. Thus two arrows and in space represent the same free vector if they have the same magnitude and direction: that is, they are equipollent if the quadrilateral ABB′A′ is a parallelogram. If the Euclidean space is equipped with a choice of origin, then a free vector is equivalent to the bound vector of the same magnitude and direction whose initial point is the origin. The term vector also has generalizations to higher dimensions, and to more formal approaches with much wider applications. Further information In classical Euclidean geometry (i.e., synthetic geometry), vectors were introduced (during the 19th century) as equivalence classes under equipollence, of ordered pairs of points; two pairs and being equipollent if the points , in this order, form a parallelogram. Such an equivalence class is called a vector, more precisely, a Euclidean vector. The equivalence class of is often denoted A Euclidean vector is thus an equivalence class of directed segments with the same magnitude (e.g., the length of the line segment ) and same direction (e.g., the direction from to ). In physics, Euclidean vectors are used to represent physical quantities that have both magnitude and direction, but are not located at a specific place, in contrast to scalars, which have no direction. For example, velocity, forces and acceleration are represented by vectors. In modern geometry, Euclidean spaces are often defined from linear algebra. More precisely, a Euclidean space is defined as a set to which is associated an inner product space of finite dimension over the reals and a group action of the additive group of which is free and transitive (See Affine space for details of this construction). The elements of are called translations. It has been proven that the two definitions of Euclidean spaces are equivalent, and that the equivalence classes under equipollence may be identified with translations. Sometimes, Euclidean vectors are considered without reference to a Euclidean space. In this case, a Euclidean vector is an element of a normed vector space