Question

The position vectors of points $A,B,C$ and $D$ are $A=3\hat{i}+4\hat{j}+5\hat{k}$, $B=4\hat{i}+5\hat{j}+6\hat{k},C=7\hat{i}+9\hat{j}+3\hat{k}$ and $D=4\hat{i}+6\hat{j}$, then the displacement vectors $\overrightarrow{AB}$ and $\overrightarrow{CD}$ are

• A. Perpendicular
• B. Parallel
• C. Antiparallel
• D. Inclined at an angle of ${60}^{\circ }$

Answer 1

The correct option is C Antiparallel

$\overrightarrow{A}=3\hat{i}+4\hat{j}+5\hat{k}$
$\overrightarrow{B}=4\hat{i}+5\hat{j}+6\hat{k}$
$\overrightarrow{C}=7\hat{i}+9\hat{j}+3\hat{k}$
$\overrightarrow{D}=4\hat{i}+6\hat{j}$
$\overrightarrow{AB}=\overrightarrow{B}-\overrightarrow{A}$
$=(4\hat{i}+5\hat{j}+5\hat{k})-(3\hat{i}+4\hat{j}+5\hat{k})$
$=\hat{i}+\hat{j}+\hat{k}$

$\overrightarrow{CD}=\overrightarrow{D}-\overrightarrow{C}$
$=(4\hat{i}+5\hat{j})-(7\hat{i}+9\hat{j}+3\hat{k})$
$=-3\hat{i}-3\hat{j}-3\hat{k}$
SInce $\overrightarrow{CD}=\lambda \left(\overrightarrow{AB}\right)(\text{where},\lambda =-3)$
Therefore , they are antiparallel.