$n$-dimensional rotation along a 2D arbitrary plane

Question

Given two vectors in $\mathbb{R}^n$, $v_0$ and $v_1$, which define a plane including the origin a rotation along that plane can be defined from $v_0$ to $v_1$. I know the formula for rotation within a plane, but I'm getting confused on how to apply that arbitrary planar rotation to the other points in $\mathbb{R}^n$. The plane itself can be oriented in any direction depending on $v_0$ and $v_1$.

Answer 1

We may assume that $v_0$ and $v_1$ are non-parallel unit vectors. Let $u\in R^n$. Then $$\langle u,v_0\rangle v_0+\langle u,v_1\rangle v_1 $$ is the projection of $u$ in the plane. In that plane we rotate $v_0$ to $v_1$ and $v_1$ to $w=2\langle v_0,v_1\rangle v_1-v_0$. (Convince yourself that $v_1=v_0+w$ and $\|w\|=1$; it's helpful to draw a picture.)

So $$u\mapsto u-\langle u,v_0\rangle v_0-\langle u,v_1\rangle v_1 +\langle u,v_0\rangle v_1+\langle u,v_1\rangle(2\langle v_0,v_1\rangle v_1-v_0) $$ $$ =u-\langle u,v_0\rangle(v_0-v_1)-\langle u,v_1\rangle(v_1-w\rangle, $$ that is, we rotate the component of $u$ in the plane and leave the rest of $u$ unchanged.

Michael