Question

Let $P,Q,R$ and $S$ be points on the plane with position vectors -$2\hat{i}-\hat{j}$, $4\hat{i},3\hat{i}+3\hat{j}$ and $-3\hat{j}+2\hat{j}$ respectively, then the quadrilateral $PQRS$ must be a

• A. Parallelogram,which is neither a rhombus nor a rectangle
• B. Square
• C. Rectangle but not a square
• D. Rhombus but not a square

Answer 1

The correct option is A Parallelogram,which is neither a rhombus nor a rectangle

Given

$\overrightarrow{P}=-2\hat{i}-\hat{j}$

$\overrightarrow{Q}=4\hat{i}$

$\overrightarrow{R}=3\hat{i}+3\hat{j}$

$\overrightarrow{S}=-3\hat{i}+2\hat{j}$

Cheking for square and rectangle

$\overrightarrow{PQ}=\overrightarrow{Q}-\overrightarrow{P}$

$\overrightarrow{PQ}=4\hat{i}+2\hat{i}+\hat{j}$

$\overrightarrow{PQ}=6\hat{i}+\hat{j}$

$\overrightarrow{PS}=\overrightarrow{S}-\overrightarrow{P}$

$\overrightarrow{PS}=-3\hat{i}+2\hat{j}+2\hat{i}+\hat{j}$

$\overrightarrow{PS}=-\hat{i}+3\hat{j}$

$\overrightarrow{PQ}\cdot \overrightarrow{PS}$ should be zero because sides make 90 angle

$\overrightarrow{PQ}\cdot \overrightarrow{PS}=-6+3=-3\ne 0$

So cannot be rectangle and square

Cheking for rhombus all sides should equal length

$\left|\overrightarrow{PQ}\right|=\sqrt{6^2+1^2}=\sqrt{37}$

$\left|\overrightarrow{PS}\right|=\sqrt{3^2+1^2}=\sqrt{10}$

$\left|\overrightarrow{PQ}\right|\ne \left|\overrightarrow{PS}\right|$

So not rhombus

to be a parallelogram opposite sides should equal

$\overrightarrow{QR}=\overrightarrow{R}-\overrightarrow{Q}$

$\overrightarrow{QR}=3\hat{i}+3\hat{j}-4\hat{i}$

$\overrightarrow{QR}=-\hat{i}+3\hat{j}$

$\overrightarrow{RS}=\overrightarrow{S}-\overrightarrow{R}$

$\overrightarrow{RS}=-3\hat{i}+2\hat{j}-3\hat{i}-3\hat{j}$

$\overrightarrow{RS}=-6\hat{i}-\hat{j}$

$\left|\overrightarrow{RS}\right|=\sqrt{6^2+1^2}=\sqrt{37}$

$\left|\overrightarrow{QR}\right|=\sqrt{3^2+1^2}=\sqrt{10}$

opposite sides are equal

$\left|\overrightarrow{PQ}\right|=\left|\overrightarrow{RS}\right|$

$\left|\overrightarrow{PS}\right|=\left|\overrightarrow{QR}\right|$

So only parallelogram