![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhsgtJkNB7coFtXTL5YfnMkbLb3y6tB0ZLPaU-OudILNRmtCJ8yWq4NEitP7UhY_gSXTQxh0iruD9dCsYGPI6rN5Nzd4jEpqoBvcN4T0wGSfLCqyxsC-JJdf35vlWMDzIxYp3T1T5c1YAGn/s320/Screen+shot+2011-06-26+at+1.41.00+PM.png)
A bit more about rotational transformations (see the previous post for the setup to this). We derived equations describing the coordinates of a point at x,y with respect to a new coordinate system in u,v that is rotated counter-clockwise by an angle θ:
We can also solve these equations for u and v. (It helps to know the answer). Notice that if we multiply the first equation by cos(θ) and the second one by sin(θ) and add, we'll end up with u by itself on the right-hand side:
Similarly, if we multiply the first equation by sin(θ) and the second one by cos(θ) and subtract the first from the second, we end up with v by itself:
I came up with geometric explanations for these relationships, the one for x and y in terms of u and v is at the top of the post, and below is the same drawing but explaining u and v in terms of x and y.
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgdpHLrZSEBy4l5roSXeCt31jCwx3jrbkwu_SpsAZrbIFaXwDMv4nP4HoNoWJD23sHnbkyd2gNGdmzSovivOnqQFwcr5YZZl-a6fqj0mCdd2XRcVMYOA10CgurMzXfZdRskLw6ePtrvm-tX/s320/Screen+shot+2011-06-26+at+1.33.51+PM.png)
A quicker way to the same place is to start by considering u and v to be the original coordinate system, and rotate clockwise through the angle θ. Then θ is negative, sin(θ) = -sin(-θ) and the cosine stays the same. We end up switching the signs of the sine terms, as we found.