Describing SO_3

[dawn mulch]

If $\varphi\in\text{SO}_3$, $\varphi(v)=-v$, and $P$ is the plane perpendicular to $v$, how do I show $\varphi$ acts like a reflection on $P$? The hint we're given is the fact $\det(\varphi)=1$, but I don't see how that plays in. I've managed to prove $w\in P\implies -\varphi(w)\in P$, but unsure if that's useful/relevant

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[wary storm]

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[dawn mulch]

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