Question

The position vectors of the points $P,Q,R,S$ are $i+j+k,2\hat{i}+5\hat{j},3\hat{i}+2\hat{j}-3\hat{k}$ and $\hat{i}-6\hat{j}-\hat{k}$ respectively. Prove that the line $PQ$ and $RS$ are parallel.

Answer 1

$\therefore$ directions of $\overrightarrow{PQ}=(2\hat{i}-\hat{i})+(5\hat{j}-\hat{j})-\hat{k}$

$=(\hat{i}+4\hat{j}-\hat{k})$

$\therefore$ directions of $\overrightarrow{RS}=(\hat{i}-3\hat{i})+(-6\hat{j}-2\hat{j})+(-\hat{k}+5\hat{k})$

$=(-2\hat{i}-8\hat{j}+2\hat{k})$

$\therefore \hat{i}+4\hat{j}-\hat{k}=\frac{-1}{2}(-2\hat{i}-8\hat{j}+2\hat{k})$

$\therefore d\left(\overrightarrow{PQ}\right)=\frac{-1}{2}\left(d\right(\overrightarrow{RS}\left)\right)$

Hence line PQ and Rs are parallel.