isometries of angles

Question

Find an isometry that maps ∠(1,1)(3,2)(2,2) to∠(3,−1)(3,2)(4,0) This is a problem for school but I don't just want the answer I want to know how to understand it. I graphed it on wolframalpha. I can clearly see that I will need to stretch it rotate and shift. But how do I know what order to do it in? My instincts tell me to rotate first but that is just a guess.

Answer 1

In fact, you can do it in either order. If you translate (shift) first, each point $\mathbf p$ will be transformed into $\mathbf p' = R(\mathbf p+\mathbf t)=R\mathbf p+R\mathbf t$, which is clearly equivalent to rotating first and then translating by $R\mathbf t$. It’s easier to figure out the correct translation if you save that for last, though, since you don’t have to “unrotate” the difference between where the rotation will put $\mathbf p$ and the desired new point $\mathbf p'$. The same holds if $R$ is a reflection rather than a rotation.