Question

If $\overline{p},\overline{q},\overline{r}$ are position vectors of the points P, Q, R respectively. Where $\overline{p}=\overline{a}+2\overline{b}+5\overline{c}$, $\overline{q}=3\overline{a}+2\overline{b}+\overline{c}$ , $\overline{r}=2\overline{a}+2\overline{b}+3\overline{c}$ , then show that the points P, Q, R are collinear.

Answer 1

$\overrightarrow{p}=\overrightarrow{a}+2\overrightarrow{b}+5\overrightarrow{c}$

$\overrightarrow{q}=3\overrightarrow{a}+2\overrightarrow{b}+\overrightarrow{c}$

$\overrightarrow{r}=2\overrightarrow{a}+2\overrightarrow{b}+3\overrightarrow{c}$

$P,Q,R$ are collinear if $\overrightarrow{p}+\overrightarrow{q}=\lambda \overrightarrow{r}$

$\Rightarrow \overrightarrow{a}+2\overrightarrow{b}+5\overrightarrow{c}+3\overrightarrow{a}+2\overrightarrow{b}+\overrightarrow{c}=\lambda (2\overrightarrow{a}+2\overrightarrow{b}+3\overrightarrow{c})$

$\Rightarrow 4\overrightarrow{a}+4\overrightarrow{b}+6\overrightarrow{c}=2\lambda \overrightarrow{a}+2\lambda \overrightarrow{b}+3\lambda \overrightarrow{c}$

$\Rightarrow 2\lambda =4,3\lambda =6$

$\therefore \lambda =2$

Hence, $P,Q,R$ are collinear.