= Solution
For <uniform density>, the <centroid> is the <volume> average of position and coincides with the <centre of mass>:
$$
\boxed{\mathbf r_G=\frac1m\int_A\rho\,\mathbf r\,dV,\qquad m=\rho\operatorname{Vol}(A).}
$$
A point fixed in the body has $\mathbf r_{OQ}=\sum_iq_i\mathbf e_i$ with time-independent body coordinates $q_i$. Since $O$ is fixed, part (a) gives
$$
\mathbf v_Q=\sum_iq_i\dot{\mathbf e}_i=\boldsymbol\omega\times\mathbf r_{OQ}.
$$
The <centroid> is fixed relative to the body as well, so $\mathbf v_G=\boldsymbol\omega\times\mathbf r_{OG}$. Using $\mathbf r_{OQ}=\mathbf r_{OG}+\mathbf r_{GQ}$ therefore gives
$$
\boxed{\mathbf v_Q=\boldsymbol\omega\times\mathbf r_{OQ}=\mathbf v_G+\boldsymbol\omega\times\mathbf r_{GQ}.}
$$
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