Question

If $\overline{p}=\hat{i}-2\hat{j}+\hat{k}$ and $\overline{q}=\hat{i}+4\hat{j}-2\hat{k}$ are position vectors of points $P$ and $Q$, find the position vector of the point $R$ which divides segment $PQ$ internally in the ratio $2:1$

Answer 1

Let $R\left(\overrightarrow{r}\right)$ is the point which divides the line segment joining the points $P$ and $Q$ internally in the ratio $2:1$.
$\overrightarrow{r}=\frac{2(\hat{i}+4\hat{j}-2\hat{k})+1(\hat{i}-2\hat{j}+\hat{k})}{2+1}$
$\overrightarrow{r}=\frac{3\hat{i}+6\hat{j}-3\hat{k}}{3}$
$\therefore \overrightarrow{r}=\hat{i}+2\hat{j}-\hat{k}$
and the coordinates of the point $R$ are $(1,2,-1)$.