Let ‘O’ be the origin and $\overline{OA}=2\overline{i}+2\overline{j}+\overline{k},\overline{OB}=\overline{i}-2\overline{j}+2\overline{k}and\overline{OC}=\frac{1}{2}(\overline{OB}-\lambda \overline{OA})$ for some $\lambda >0$ if $|\overline{OB}\times \overline{OA}|=\frac{9}{2}$ then which of the following statements is/are true

$\overline{OB}\times \overline{OC}=\frac{1}{2}(\overline{OB}\times \overline{OB}-\lambda \overline{OA})=\frac{\lambda }{2}(\overline{OA}\times \overline{OB})$.

Since $\overline{OA}.\overline{OB}=0\Rightarrow \overline{OA}\perp \overline{OB}$
$\therefore |\overline{OB}\times \overline{OC}|=\left|\frac{\lambda }{2}(\overline{OA}\times \overline{OB})\right|=\frac{\left|\lambda \right|}{2}\left|\overline{OA}\right|\left|\overline{OB}\right|\sin \frac{\pi }{2}$
$\Rightarrow \frac{9}{2}=\frac{\left|\lambda \right|}{2}\mathrm{.3.3.1}\Rightarrow \lambda =1(\therefore \lambda >0)$
$\overline{OC}=\frac{\overline{OB}-\overline{OA}}{2}=\frac{1}{2}(-\overline{i}-4\overline{j}+\overline{k})$

a) Projection of $\overline{OC}on\overline{OA}=\frac{\overline{OC}.\overline{OA}}{\left|\overline{OA}\right|}=\frac{-3}{2}$
b) Area of all $OAB=\frac{1}{2}|\overline{OA}\times \overline{OB}|=\frac{9}{2}$
c) Area of all $ABC=\frac{1}{2}|\overline{AB}\times \overline{AC}|=\frac{9}{2}$
d)$\cos \theta =\frac{(\overline{OA}+\overline{OC}).(\overline{OA}-\overline{OC})}{|\overline{OA}+\overline{OC}||\overline{OA}-\overline{OC}|}=\frac{18}{3\sqrt{2}\sqrt{90}}\Rightarrow \theta \pm \frac{\pi }{3}$