Let \( O \) be the origin and \( \overrightarrow{O A}=2 \hat{i}+2 \hat{j}+\hat{k}, \overrightarr...

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Let \( O \) be the origin and \( \overrightarrow{O A}=2 \hat{i}+2 \hat{j}+\hat{k}, \overrightarrow{O B}=\hat{i}-2 \hat{j}+2 \hat{k} \) and \( \overrightarrow{O C}=\frac{1}{2}(O B-\lambda \overrightarrow{O A}) \) for some \( \lambda0 \). If \( |\overrightarrow{O B} \times \overrightarrow{O A}|=\frac{9}{2} \), then which of the following statements is (are) TRUE?
(a) Projection of \( \overrightarrow{O C} \) on \( \overrightarrow{O A} \) is \( -\frac{3}{2} \)
(b) Area of the triangle \( O A B \) is \( \frac{9}{2} \)
(c) Area of the triangle \( A B C \) is \( \frac{9}{2} \)
(d) The acute angle between the diagonals of the parallelogram with adjacent sides \( \overrightarrow{O A} \) and \( \overrightarrow{O C} \) is \( \frac{\pi}{3} \).
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