Rotating Rigid Body

We need $\mathbf{\omega} \times \left(\begin{array}{c}1 \\ 2 \\ 2 \end{array}\right) \; = \; \left(\begin{array}{c} 4 \\ -4 \\ 2 \end{array}\right) \hspace{2cm} \mathbf{\omega} \times \left(\begin{array}{c} 0 \\ 3 \\ 3 \end{array}\right) \; = \; \left(\begin{array}{c} 6 \\ -4 \\ 4 \end{array}\right)$ and hence $\omega \times \left(\begin{array}{c}1 \\ 0 \\ 0 \end{array}\right) \; = \; \left(\begin{array}{c} 0 \\ -\frac43 \\ -\frac23 \end{array}\right) \hspace{2cm} \omega \times \left(\begin{array}{c} 0 \\ 1 \\ 1 \end{array}\right) \; = \; \left(\begin{array}{c} 2 \\ -\frac43 \\ \frac43 \end{array}\right)$ Thus $\left(\begin{array}{c} 0 \\ \omega_2 \\ \omega_3\end{array}\right) \; = \; \omega - \left(\omega \cdot \mathbf{i}\right) \mathbf{i} \; = \; \mathbf{i} \times (\mathbf{\omega} \times \mathbf{i}) \; = \; \left(\begin{array}{c} 1 \\ 0 \\ 0 \end{array}\right) \times \left(\begin{array}{c} 0 \\ -\frac43 \\ -\frac23 \end{array}\right) \; = \; \left( \begin{array}{c} 0 \\ \frac23 \\ -\frac43 \end{array}\right)$ and hence $\left(\begin{array}{c} 2 \\ -\frac43 \\ \frac43 \end{array}\right) \; = \; \mathbf{\omega} \times \left(\begin{array}{c} 0 \\ 1 \\ 1 \end{array}\right) \; = \; \left(\begin{array}{c} \omega_1 \\ \frac23 \\ -\frac43 \end{array}\right) \times \left(\begin{array}{c} 0 \\ 1 \\ 1 \end{array}\right) \; = \; \left(\begin{array}{c} 2 \\ -\omega_1 \\ \omega_1 \end{array}\right)$ so that $\omega_1 = \frac43$ and hence $\mathbf{\omega} \; = \; \left(\begin{array}{c} \frac43 \\ \frac23 \\ -\frac43 \end{array} \right)$ which has magnitude $\boxed{2}$ .