(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

An (x,y) coordinate system is rotated through an angle theta to produce an (x',y') system, see figure.

A point with coordinates (x,y) will have coordinates (x',y') in the rotated system given by:

x'_{1}= (x_{1}* cos theta) + (y_{1}* sin theta)

y'_{1}= (-x_{1}* sin theta) + (y_{1}* cos theta)

Show that the formula for the distance of the point from the origin is invariant, or unchanged, by the rotation. That is, show:

sqrt (x_{1}^{2}+ y_{1}^{2}) = sqrt (x'_{1}^{2}+ y'_{1}^{2})

2. Relevant equations

I don't know if these are really relevant, I just thought so:

a_{x}= a cos theta

a_{y}= a sin theta

a = sqrt (a_{x}^{2}+ a_{y}^{2})

tan theta = a_{y}/a_{x}

where a = the magnitude of vector a and theta = the angle vector a makes with the positive direction of the x axis

3. The attempt at a solution

So I thought this was asking, more or less, to prove that rotating the axes changes the components of the vector but not the vector itself.

I set the two equations given equal to each other, subbing in the information given for x' and y', but I don't know how to proceed or even if this was a good place to start. Any ideas for starting off?

sqrt (x_{1}^{2}+ y_{1}^{2}) = sqrt (((x_{1}* cos theta) + (y_{1}* sin theta))^{2}+ ((-x_{1}* sin theta) + (y_{1}* cos theta))^{2})

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# Homework Help: Vector Proof

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