Vector Algebra: Let O be the origin and overrightarrowOA=2hati+2hatj+hatk ,… | JEE Advanced 2021 Paper 2

JEE Advanced 2021 Paper 2Question 1

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 $\left|\overrightarrow{OB}\times\overrightarrow{OC}\right|=\frac{9}{2}$, then which of the following statements is (are) TRUE ?

Vector Algebra

More than one option may be correct.

Projection of $\overrightarrow{OC}$ on $\overrightarrow{OA}$ is $-\frac{3}{2}$✓ Correct answerArea of the triangle $OAB$ is $\frac{9}{2}$✓ Correct answerArea of the triangle $ABC$ is $\frac{9}{2}$✓ Correct answerThe acute angle between the diagonals of the parallelogram with adjacent sides $\overrightarrow{OA}$ and $\overrightarrow{OC}$ is $\frac{\pi}{3}$

Solution

$|\vec{OA}\times\vec{OB}|=9$ and $|\vec{OB}\times\vec{OC}|=\frac\lambda2\cdot9=\frac92$ gives $\lambda=1$, $\vec{OC}=\frac12(-1,-4,1)$. Projection of $OC$ on $OA$ is $\frac{-9/2}{3}=-\frac32$; areas of $OAB$ and $ABC$ are both $\frac92$; the diagonals $OA\pm OC$ give $\cos\theta=\frac{4.5}{\sqrt{4.5}\sqrt{22.5}}\ne\frac12$, so (D) fails.

Vector Algebra - JEE Mathematics question (JEE Advanced 2021 Paper 2)

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