If $$(x,y)$$ and $$(X,Y)$$ are the coordinates of the same point referred yo ywo sets of rectangular axes with the same origin and if $$ux+vy$$, where $$u$$ and $$v$$ are indepedent of $$x$$ and $$y$$, becomes $$VX+UY$$, show that $$u^ {2}+v^ {2}=U^ {2}+V^ {2}$$