I'm currently reading the book "Finite Reflection Groups" by Grove and Benson for a kind of undergraduate course the Germans call "Proseminar" - it basically means I have to read up on a topic which to then give a presentation on. However, one particular paragraph in this book gets me stumped - I'm not sure whether I've stumbled upon a mistake in the book or whether I'm mistaken about this. On page 11, the last paragraph reads as follows:
Suppose that $\dim V=3$ and that $W$ is a plane in $V$, i.e., a subspace of dimension $2$. If $R$ is a rotation in $\mathcal O(W)$, then $R$ may be extended to a rotation in $\mathcal O(V)$ if we set $Rx=x$ for all $x\in W^\perp$ and extend by linearity. If a basis $\{x_1,x_2,x_3\}$ is chosen for $V$, with $x_1\in W^\perp,x_2,x_3\in W$, then the matrix representing $R$ is
$A=\begin{bmatrix}1&0&0\\0&\cos\theta&-\sin\theta\\0&\sin\theta&\cos\theta\end{bmatrix}$.
So my question is - do not we have to assume that the aforementioned basis is an orthonormal basis? (Of course, from the definition, $x_1$ is perpendicular to the other two vectors but I don't see why we need not assume that these are unit vectors and that $x_2\perp x_3$?) Hope, someone can point me in the right way if there's a detail I'm too blind to see.