Answer
$$
\begin{aligned}
& \omega=\{2 \mathbf{i}+42.4 \mathbf{j}+43.4 \mathbf{k}\} \mathrm{rad} / \mathrm{s} \\
& \boldsymbol{\alpha}=\{-42.4 \mathbf{i}-82.9 \mathbf{j}+84.9 \mathbf{k}\} \mathrm{rad} / \mathrm{s}^2
\end{aligned}
$$
Work Step by Step
$$
\begin{aligned}
\omega & =\omega_z+\omega_s+\omega_x \\
& =1 \mathbf{k}+60 \cos 45^{\circ} \mathbf{j}+60 \sin 45^{\circ} \mathbf{k}+2 \mathbf{i} \\
& =2 \mathbf{i}+42.426 \mathbf{j}+43.426 \mathbf{k} \\
& =\{2 \mathbf{i}+42.4 \mathbf{j}+43.4 \mathbf{k}\} \mathrm{rad} / \mathrm{s} \\
\dot{\omega} & =\dot{\omega}_z+\dot{\omega}_s+\dot{\omega}_x \\
& =0+\left(\omega_z+\omega_x\right) \times \omega_s+\omega_z \times \omega_x \\
& =0+(1 \mathbf{k}+2 \mathbf{i}) \times(42.426 \mathbf{j}+43.426 \mathbf{k})+1 \mathbf{k} \times(2 \mathbf{i}) \\
& =-42.426 \mathbf{i}+84.853 \mathbf{k}-84.853 \mathbf{j}+2 \mathbf{j} \\
& =\{-42.4 \mathbf{i}-82.9 \mathbf{j}+84.9 \mathbf{k}\} \mathrm{rad} / \mathrm{s}^2
\end{aligned}
$$
