For φ in [0, 2pi) and (x_{1, }x_{2}) in R^{2 }we define a transformation.

R(φ):R^{2 }---> R^{2}

x ---> R(φ)x=y=(y_{1},y_{2})

where

y_{1}=x_{1}cosφ+x_{2}sinφ

y_{2}=-x_{1}sinφ+x_{2}cosφ

Prove ||y||=||R(φ)x||=||x||

ie. R(φ) leaves the euclidean norm invariant.

Now geometry isn't my strong point to start with but I usually grasp the topics enough to do the homeworks well, but this may as well be in dutch for as much as I can figure out what to do. We haven't done anything even remotely similar in class and I'm at a loss as to where to even start. I would be really really grateful if someone could help me out a little. Thanks