Rotations of vectors on the plane can be seen either as active or passive. In the former we rotate the vector by an angle $\theta$ whereas in the latter we rotate the coordinate basis by $-\theta$. We end up with the same components $(a,b)$ for the rotated vector.
The group of rotations on the plane $SO(2)$ excludes reflections about an axis. For example, from a passive point of view, the reflection $S(y):(\hat i,\hat j)\rightarrow (-\hat i,\hat j)$ reflects the $x$ axis about the $y$ axis. The vector $\vec v=(1,1)^T$ would then be written as $\vec v'=(-1,1)^T$. The point is that there is no passive rotation able to do that, so $s$ does not belong to $SO(2)$.
Then it comes my doubts. Let us adopt the active point of view. The equivalent rotated vector would be $\vec v'=(-1,1)^T$ but then this is simply an active rotation by $\pi/2$ which in fact belongs to $SO(2)$.
Perhaps my questions are very basic: Is my assumption on what an active reflection about an axis is correct? If so, then why does every possible active reflection seems to be an element of $SO(2)$ whereas the same does not happen for the passive point of view.
Edit: As a concrete example: Take an arbitrary vector $(a,b)^T$. The reflected (about $y$) vector is $(-a,b)^T$. However this can also be obtained by an active rotation, namely $R(\pi-2\arctan(b/a))$ which has determinant one