= Solution
In the massive particle's rest frame, $P=(m_0c,\mathbf0)$ and the physical <Pauli-Lubanski pseudovector> is $W=(0,m_0c\,\mathbf s_{\rm rest})$, up to a common overall sign. Their <Minkowski inner product> is therefore zero. Since it is a <Lorentz scalar>,
$$
\boxed{W\cdot P=0,\qquad U_W\cdot P=0}
$$
in every inertial frame, for any nonzero scalar normalization of $W$. This proves the requested orthogonality without a boosted component calculation.
The printed assertion that its particular normalization is always a unit vector needs qualification. For the displayed components $W=(\mathbf s\cdot\mathbf p,E\mathbf s/c)$,
$$
W^2=-m_0^2c^2|\mathbf s|^2-|\mathbf s\times\mathbf p|^2.
$$
Thus dividing by $m_0c|\mathbf s|$ is unit spacelike only when $\mathbf s\times\mathbf p=0$, or when $\mathbf s$ in that denominator is explicitly replaced by the proper rest-spin magnitude with correctly boosted components of $W$. As a counterexample to the literal displayed normalization, choose $\mathbf s\perp\mathbf p$ with nonzero momentum: it gives $U_W^2=-\gamma^2$, not $-1$. Orthogonality remains true, but it does not establish normalization.
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