Rectangular Coordinate System

The following discussion is limited to vectors in a two‐dimensional coordinate plane, although the concepts can be extended to higher dimensions.

If vector  is shifted so that its initial point is at the origin of the rectangular coordinate plane, it is said to be in standard position. If vector  is equal to vector  and has its initial point at the origin, it is said to be the standard vector for . Other names for the standard vector include radius vector and position vector (Figure  1).

Figure 1 
              Vectors drawn on a plane.

Vector  is the standard vector for all vectors in the plane with the same direction and magnitude as . In order to find the standard vector for a geometric vector in the coordinate plane, only the coordinates of point P must be found because point 0 is at the origin. If the coordinates of point A are ( x _a, y _a ) and the coordinates of point B are ( x _b, y _b), then the coordinates of point P are ( x _b − x _a, y _ab − y _a).

Example 1: If the endpoints of a vector  have coordinates of A(−2, −7) and B (3, 2), then what are the coordinates of point P such that  is a standard vector and  =  (see Figure  2)?