Editing Euclidean vector (section)

===In Cartesian space===
In the [[Cartesian coordinate system]], a bound vector can be represented by identifying the coordinates of its initial and terminal point. For instance, the points {{math|''A'' {{=}} (1, 0, 0)}} and {{math|''B'' {{=}} (0, 1, 0)}} in space determine the bound vector <math>\overrightarrow{AB}</math> pointing from the point {{math|''x'' {{=}} 1}} on the ''x''-axis to the point {{math|''y'' {{=}} 1}} on the ''y''-axis.

In Cartesian coordinates, a free vector may be thought of in terms of a corresponding bound vector, in this sense, whose initial point has the coordinates of the origin {{math|''O'' {{=}} (0, 0, 0)}}. It is then determined by the coordinates of that bound vector's terminal point. Thus the free vector represented by (1, 0, 0) is a vector of unit length—pointing along the direction of the positive ''x''-axis.

This coordinate representation of free vectors allows their algebraic features to be expressed in a convenient numerical fashion. For example, the sum of the two (free) vectors (1, 2, 3) and (−2, 0, 4) is the (free) vector
<math display=block>(1, 2, 3) + (-2, 0, 4) = (1-2, 2+0, 3+4) = (-1, 2, 7)\,.</math>