How to rotate any plane in 3d to (XY) plane?

Question

I want to rotate a given plane in 3d to (XY) so i can work like in normal 2d. Because if they ask like what is the equation of a circle on a given plane, i'll just rotate it to (XY) plane and apply the normal 2d formula. But i'm in final year of highschool and my teacher said that you don't need to make that complicated, but i want to know even if i didn't use it in a test. so if can anyone help me about it. P.S: can you wrap this topic up in terms of cross product, dot product or some basic stuff not like rotation matrix or so, if you have to do it can you explain really well because like i said i'm in final year in highschool not college. thanks in advance

Answer 1

A generic plane $\prod\in \mathbb{R}^3$ has a normal unit vector $\hat n$ so we need a rotation matrix $R(\theta,\hat u)$ which transforms $\hat n$ into $\hat e_z$. Taking $L$ as the intersection line between $\prod$ and the $XY$ plane, we can use Rodrigues formula around the line $L$ with direction $\hat u$ to perform a rotation of $\theta=\arccos\left(\hat n\cdot \hat e_z\right)$ Rodrigues formula