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 and Try computing and individually, then add them together and see if you can get The trig. identity 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).