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 $△OAB=9/2$
(C) Area of $△ABC=9/2$
(D) $\overrightarrow{OA}+\overrightarrow{OC}=\frac{3\hat{i}+3\hat{x}}{2}$,
$\overrightarrow{OA}-\overrightarrow{OC}=\frac{5\hat{i}+8\hat{j}+\hat{k}}{2}$
$(\overrightarrow{OA}+\overrightarrow{OC})(\overrightarrow{OA}-\overrightarrow{OC})=|\overrightarrow{OA}+\overrightarrow{OC}||\overrightarrow{OA}-\overrightarrow{OC}|\cos \theta$
$\cos \theta =\frac{1}{\sqrt{5}}$