Showing reflection in a plane containing origin is represented by symmetric orthogonal matrix I'm sure this question is pretty straightforward but I've been scratching my head at this for a small while 

Assuming that any isometry $T: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ may be written in the form $T({\bf{v}}) = P{\bf{v}} + {\bf{q}}$ where ${\bf{q}} \in \mathbb{R}^3$ and $P$ is a $3 \times 3$ orthogonal matrix show that a reflection in a plane containing the origin is represented by a symmetric orthogonal matrix. 

I've been trying to use the fact that if we've got some ${\bf{v}} \in \mathbb{R}^3$ then we can write reflection as $T({\bf{v}}) = P{\bf{v}}$ (for some reason I feel like the constant vector isn't needed for reflection) - then since we're referring to reflection we must have 
$TT({\bf{v}}) = T^2({\bf{v}}) = P^2{\bf{v}}= I{\bf{v}}$ (essentially saying reflection twice will give us the original vector)
We know that $P$ is orthogonal, so $P^TP = I$, then using $P^2 = I$ we get $P^TP = P^2 \implies P^T = P$ hence it is a symmetric orthogonal matrix.
Is this correct? Thanks for any help.

Looks ok to me. Vector $q$ equals $0$ because $T(0)=P(0)+q$ i.e. $0=0+q$.
