Question

The figure formed by the four points $\hat{i}+\hat{j}-\hat{k},2\hat{i}+3\hat{j},5\hat{j}-2\hat{k}$ and $\hat{k}-\hat{j}$ is

• A. Trapezium
• B. Rectangle
• C. Parallelogram
• D. None of the above

Answer 1

Answer: (D)

None of the above

Let $\overrightarrow{P}=\hat{i}+\hat{j}-\hat{k},\overrightarrow{Q}=2\hat{i}+3\hat{j},\overrightarrow{R}=5\hat{j}-2\hat{k}$ and $\overrightarrow{S}=-\hat{j}+\hat{k}$
$\therefore \overrightarrow{PQ}=\hat{i}+2\hat{j}+\hat{k}$
$\Rightarrow \left|\overrightarrow{PQ}\right|=\sqrt{1^2+2^2+1^2}=\sqrt{6}$
and $\overrightarrow{QR}=-2\hat{i}+2\hat{j}-2\hat{k}$
$\Rightarrow \left|\overrightarrow{QR}\right|=\sqrt{{(-2)}^2+2^2+{(-2)}^2}=\sqrt{12}$
and $\overrightarrow{RS}=-6\hat{j}+3\hat{k}\Rightarrow \left|\overrightarrow{RS}\right|=\sqrt{36+9}=\sqrt{45}$
Similarly, $\overrightarrow{SP}=\hat{i}+2\hat{j}-2\hat{k}$
$\Rightarrow \left|\overrightarrow{SP}\right|=\sqrt{1^2+2^2+{(-2)}^2}=\sqrt{9}=3$
Which are not satisfied the condition of any of the trapezium, rectangle and parallelogram.