Solution
= Solution
The <matrix> is
$$
P=I-\mathbf p\mathbf p^T=\begin{pmatrix}0&-1&0\\-1&0&0\\0&0&1\end{pmatrix},
$$
so
$$
\boxed{P\mathbf x=(-x_2,-x_1,x_3)^T.}
$$
Here $\|\mathbf p\|^2=2$, so $P=I-2\hat{\mathbf p}\hat{\mathbf p}^T$ with $\hat{\mathbf p}=\mathbf p/\sqrt2$. This is a <Householder transformation>: \b[reflection in the <plane> $x_1+x_2=0$]. The component parallel to the <normal vector> $\mathbf p$ changes sign, while the perpendicular component is fixed.