# Thread: Prove the plane transformation is the rotation

1. ## Prove the plane transformation is the rotation

Let G be an angle. Show that the plane transformation Q=f(P) that maps the point P(x,y) to the point Q(x',y') where:

x' = x * cos(G) - y*sin(G)
y' = x* sin(G) + y*cos(G)

is in fact the rotation Ro,G.

I understand the meaning of G,Q,P, and the function. However, I don't know what to do with the x' and y'. I don't know how to start this proof. What is the strategy to do it? and how to do it?

2. The rotation about (0,0) through angle G has three properties: (0, 0) is mapped into itself. The distance from (x,y) to (0, 0) is the same as the distance form Q(x,y) to (0, 0). The angle formed by the rays from (0,0) to (x,y) and to (x',y') is the same as the angle formed by the rays from (0,0) to Q(x,y) and to Q(x',y'). Show those three things.

For example, it is easy to show that x'= 0(cos G)- 0(sin G)= 0 and y'= 0(sin G)+ 0(cos G)= 0.

The distance from (0, 0) to Q(x, y) is $\sqrt{(xcos G- ysin G)^2+ (x sin G + ycos G)^2}$ $= \sqrt{x^2 cos^2 G- 2xy sin G cos G+ y^2 sin^2 G+ x^2 sin^2 G+ 2xy sin G cos G+ y^2 cos^2 G}= ?$