For each \(\alpha \in \mathbb{R}\) and \(\omega, ho \in \mathbb{R}^{3}\) with \(\|ho\| \leq 1\), deduce that the matrix-valued function

\[
\begin{aligned}
M_{v}\left(\left\|x-x^{\prime}ight\|ight) M_{v}\left(t-t^{\prime}ight) \times & \left(\cos \left(\omega^{\prime}\left(x-x^{\prime}ight)+\alpha\left(t-t^{\prime}ight)ight) I_{3}ight. \\
& \left.-\sin \left(\omega^{\prime}\left(x-x^{\prime}ight)-\alpha\left(t-t^{\prime}ight)ight) \chi(ho)ight)
\end{aligned}
\]

is the covariance function for a Gaussian process \(Z: \mathbb{R}^{3} \times \mathbb{R} ightarrow \mathbb{R}^{3}\) that is both temporally and spatially stationary.