transformation - sorry really don't know how to describe the question!

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 :D

Re: transformation - sorry really don't know how to describe the question!

Hi carla1985,

I think the notation of this problem may be making things more confusing than anything else. Remember that $\displaystyle \|x\|=\sqrt{x_{1}^{2}+x_{2}^{2}}$ and $\displaystyle \|y\|=\sqrt{y_{1}^{2}+y_{2}^{2}}.$ Try computing $\displaystyle y_{1}^{2}$ and $\displaystyle y_{2}^{2}$ individually, then add them together and see if you can get $\displaystyle x_{1}^{2}+x_{2}^{2}.$ The trig. identity $\displaystyle \cos^{2}\varphi+\sin^{2}\varphi=1$ will be useful when doing this.

Give it another shot. If you're still stuck let me know. Good luck!

Re: transformation - sorry really don't know how to describe the question!

I agree, the notation is confusing, its still all qute new and is a lot to take in lol.

I'l have another go at it now, using your suggestions and see how it goes. Thank you :)

Re: transformation - sorry really don't know how to describe the question!

here is what is happening:

R(φ) takes a point x = (x_{1},x_{2}) to another point y = (y_{1},y_{2}).

we want to show if |x| = a, that |R(φ)(x)| = a, as well.

what do we mean by |x|? we mean: √[(x_{1})^{2} + (x_{2})^{2}]

(this is just the pythagorean theorem in disguise).

so now we have to calculate |R(φ)(x)| = |y|.

this is going to be √[(y_{1})^{2} + (y_{2})^{2}].

the algebra is going to get messy, here is where we start:

|R(φ)(x)| = √[(y_{1})^{2} + (y_{2})^{2}]

= √[(x_{1}cosφ + x_{2}sinφ)^{2} + (-x_{1}sinφ + x_{2}cosφ)^{2}]

= √[(x_{1})^{2}cos^{2}φ + 2x_{1}x_{2}sinφcosφ + (x_{2})^{2}sin^{2}φ + (x_{1})^{2}sin^{2}φ - 2x_{1}x_{2}sinφcosφ + (x_{2})^{2}cos^{2}φ]

i've done the hard part, you just have to simplify.

(something which doesn't help with the calculation but makes what R(φ) "make sense" is this: R(φ) represents what a rotation through the angle φ around the origin does to the point x. |x| represents "how far it is from x to the origin". so the upshot of all of this is that you are proving: "rotations about the origin preserve distances from the origin", something that is intuitively obvious).