Question

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

• A. Projection of $\overrightarrow{ OC }$ on $\overrightarrow{ OA }$ 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 sides $\overrightarrow{ OA }$ and $\overrightarrow{ OC }$ is $\frac{\pi}{3}$

Answer 1

Answer: (A), (B), (C)

Projection of $\overrightarrow{ OC }$ on $\overrightarrow{ OA }$ is $-\frac{3}{2}$

Answer 2

(A) Projection of $\overrightarrow{ OC }$ on $\overrightarrow{ OA }$ is $-\frac{3}{2}$
(B) Area of the triangle $OAB$ is $\frac{9}{2}$
(C) Area of the triangle $ABC$ is $\frac{9}{2}$