Let O be the origin and $\overrightarrow{\mathrm{OA}}=2 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}}, \overrightarrow{\mathrm{OB}}=\hat{\mathrm{i}}-2 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{OC}}=\frac{1}{2}(\overrightarrow{\mathrm{OB}}-\lambda \overrightarrow{\mathrm{OA}})$ for some $\lambda>0$. If $|\overrightarrow{\mathrm{OB}} \times \overrightarrow{\mathrm{OC}}|=\frac{9}{2}$, then which of the following statements is (are) TRUE?