Inner-product question Let $V$ be $\mathbf{R}^2$ equipped with usual inner product, and $v$ be a nonzero vector. $S_v(u)= u- 2 \frac{\langle u,v\rangle}{\langle v,v\rangle } v$ and $\Phi$ be a non-empty set of unit vectors in $\mathbf{R}^2$ such that $S_v(u) \in \Phi$ and $2\langle u,v\rangle\in\mathbf{Z}$ for any $u,v \in \Phi$.
I need to show that $|\Phi|=2,4$ or $6$ and describe the possible sets $\Phi$ geometrically.
I can see that if $v$ is in $\Phi$, then so is $-v$, and that $S_v(u)$ is the reflection of $u$ in the line orthogonal to $v$. I've tried taking an orthonormal basis, but that just seems to confirm the $2\langle u,v\rangle \in \mathbf{Z}$ statement. Anyone have any ideas?

First, note that if $u,v \in \mathbb{S}^1$, so is S_v(u).
Then $2<u,v> \in \mathbb{Z}$ means $2cos(\theta) \in \mathbb{Z}$, where theta is the angle between $u$ and $v$. Then $\theta=0, \pi/3,\pi/2$ or $\pi$, and you 're done!!
