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