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?
Select ALL correct options:
A
Projection of $\overrightarrow{\mathrm{OC}}$ on $\overrightarrow{\mathrm{OA}}$ is $-\frac{3}{2}$
B
Area of the triangle OAB is $\frac{9}{2}$
C
Area of the triangle ABC is $\frac{9}{2}$
D
The acute angle between the diagonals of the parallelogram with adjacent side $\overrightarrow{\mathrm{OA}}$ and $\overrightarrow{\mathrm{OC}}$ is $\frac{\pi}{3}$
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