If in the plane with the given and axes new coordinate axes are chosen parallel to the given ones, we say that there has been a translation of axes in the plane.
Let be any point in the -plane. Let be the fixed point in the - plane. We draw two perpendicular axes through : the -axis is parallel to the -axis and the -axis parallel to the -axis, as shown in the given diagram. In fact, is the origin of the new -plane. The point has the coordinates with respect to the -plane.
The equations , are called transformation equations and are used to find the coordinates of a point with respect to the new coordinate system, the -system. Thus, the point with respect to the XY-plane is .
Conversely, if the coordinates of a point with respect to the -system are given, then the coordinates with respect to the original system can be determined by the equations , .
Example 1: Let and be two points in the XY-coordinates system. Find the XY-coordinates of referring to the translated axes and .
Solution: Here and . The coordinates of referring to the new -coordinates system are
Example 2: Let be a point referring to the -coordinate system translated thorough . Find the coordinates of referring to the original coordinate system, the -system.
Solution: Here and . The coordinates of referring to the new XY-coordinates system are