Let $O$ be the origin and $\overrightarrow { O A } = 2 \hat { \mathrm { i } } + 2 \hat { \mathrm { j } } + \hat { \mathrm { k } } , \quad \overrightarrow { O B } = \hat { \mathrm { i } } - 2 \hat { \mathrm { j } } + 2 \hat { \mathrm { k } }$ and $\overrightarrow { O C } = \frac { 1 } { 2 } ( \overrightarrow { O B } - \lambda \overrightarrow { O A } )$ for some $\lambda > 0$. If $| \overrightarrow { O B } \times \overrightarrow { O C } | = \frac { 9 } { 2 }$, 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 $\frac { \pi } { 3 }$
Let $O$ be the origin and $\overrightarrow { O A } = 2 \hat { \mathrm { i } } + 2 \hat { \mathrm { j } } + \hat { \mathrm { k } } , \quad \overrightarrow { O B } = \hat { \mathrm { i } } - 2 \hat { \mathrm { j } } + 2 \hat { \mathrm { k } }$ and $\overrightarrow { O C } = \frac { 1 } { 2 } ( \overrightarrow { O B } - \lambda \overrightarrow { O A } )$ for some $\lambda > 0$. If $| \overrightarrow { O B } \times \overrightarrow { O C } | = \frac { 9 } { 2 }$, 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 $\frac { \pi } { 3 }$