If the points $P, Q$ and $R$ are with the position vectors $\hat{i}-2 \hat{j}+3 \hat{k},-2 \hat{i}+3…
If the points $P, Q$ and $R$ are with the position vectors $\hat{i}-2 \hat{j}+3 \hat{k},-2 \hat{i}+3 \hat{j}+2 \hat{k}$ and $-8 \hat{i}+13 \hat{j}$ respectively, then these points are
collinear and Q lies between P and R .
collinear and R lies between P and Q .
collinear and P lies between Q and R .
non-collinear.
Solution
$\begin{aligned}
\overline{P Q} & =-2 \hat{i}+3 \hat{j}+2 \hat{k}-(\hat{i}-2 \hat{j}+3 \hat{k}) \\
& =-3 \hat{i}+5 \hat{j}-\hat{k} \\
\overline{Q R} & =-8 \hat{i}+13 \hat{j}-(-2 \hat{i}+3 \hat{j}+2 \hat{k}) \\
& =-6 \hat{i}+10 \hat{j}-2 \hat{k} \\
& =2(-3 \hat{i}+5 \hat{j}-\hat{k})=2 \overline{P Q}
\end{aligned}$
$\therefore \quad \overline{\mathrm{QR}}$ is a scalar multiple of $\overline{\mathrm{PQ}}$.
$\therefore \quad \overline{\mathrm{QR}}$ and $\overline{\mathrm{PQ}}$ are parallel to each other with point Q in common.
$\therefore \quad$ Points $\mathrm{P}, \mathrm{Q}$ and R are collinear and Q lies between P and R .