In the euclidian field $R^3$, how many isometries apply $(1,0,0)$ on $(\sqrt{2}/2,\sqrt{2}/2,0)$ and $(0,1,0)$ on $(0,0,1)$ ?
I am tempted to answer only one, the one which apply $(0,0,1)$ on $(-\sqrt{2}/2,\sqrt{2}/2,0)$ but I feel like there should be more.
If you talk about isomotries of the metric space, there are infinitely many. If you talk about linear isometries, there are two. If you talk about linear orientation-preserving isomoetries, then there is only one.