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)

$\overrightarrow{ OA } \cdot \overrightarrow{ OB }=0 \Rightarrow \overrightarrow{ OA } \perp \overrightarrow{ OB }$
$\overrightarrow{ OC }=\frac{1}{2}((1-2 \lambda) \hat{ i }+(-2-2 \lambda) \hat{ j }+(2-\lambda) \hat{ k })$
$|\overrightarrow{ OB } \times \overrightarrow{ OC }|=\frac{9|\lambda|}{2}=\frac{9}{2} $
$\Rightarrow \lambda=\pm 1, $ as $ \lambda>0, \lambda=1 $
$\overrightarrow{ OC }=\frac{\overrightarrow{ OB }-\overrightarrow{ OA }}{2}$
(A) Projection $\overrightarrow{ OC }$ on $\overrightarrow{ OA }$
$\overrightarrow{ OC } \cdot \widehat{ OA }=-\frac{3}{2}$
(B) Area of $\triangle OAB =9 / 2$
(C) Area of $\triangle ABC =9 / 2$
(D) $\overrightarrow{ OA }+\overrightarrow{ OC }=\frac{3 \hat{i}+3 \hat{x}}{2}$,
$ \overrightarrow{ OA }-\overrightarrow{ OC }=\frac{5 \hat{i}+8 \hat{j}+\hat{k}}{2}$
$(\overrightarrow{ OA }+\overrightarrow{ OC })(\overrightarrow{ OA }-\overrightarrow{ OC })=|\overrightarrow{ OA }+\overrightarrow{ OC }||\overrightarrow{ OA }-\overrightarrow{ OC }| \cos \theta $
$\cos \theta=\frac{1}{\sqrt{5}}$