Question

The position vectors of four points A,B,C and D are$\hat{i}+\hat{j}+\hat{k},2\hat{i}+5\hat{j},3\hat{i}+2\hat{j}-3\hat{k},\hat{i}-6\hat{j}-\hat{k}$ respectively. Then select the correct relation.

• A. $AB=CD$
• B. $AB=\frac{1}{2}CD$
• C. $AB=\frac{2}{3}CD$
• D. $AB=\frac{1}{4}CD$

Answer 1

The correct option is B $AB=\frac{1}{2}CD$

$\overrightarrow{A}=\hat{i}+\hat{j}+\hat{k}$
$\overrightarrow{B}=2\hat{i}+5\hat{j}$
$\overrightarrow{C}=3\hat{i}+2\hat{j}-3\hat{k}$
$\overrightarrow{D}=\hat{i}+6\hat{j}-\hat{k}$
$\overrightarrow{AB}=\overrightarrow{B}-\overrightarrow{A}$
$\overrightarrow{AB}=\hat{i}+4\hat{j}-\hat{k}$
$\Rightarrow AB=\sqrt{1+16+1}=\sqrt{18}$
$\overrightarrow{CD}=\overrightarrow{D}-\overrightarrow{C}$
$\overrightarrow{CD}=-2\hat{i}-8\hat{j}+2\hat{k}$
$CD=\sqrt{4+64+4}=\sqrt{72}=2\sqrt{18}$
$\therefore AB=\frac{1}{2}CD$