Find the position vector of a point $R$ which divides the line joining the two points $P$ and $Q$ with position vectors $\overrightarrow{OP} = 2\vec{a} + \vec{b}$ and $\overrightarrow{OQ} = \vec{a} - 2\vec{b}$ respectively,in the ratio $1:2$,$(i)$ internally and (ii) externally.

(A) $(i)$ The position vector of the point $R$ dividing the line segment joining $P$ and $Q$ internally in the ratio $m:n = 1:2$ is given by the section formula: $\overrightarrow{OR} = \frac{m\overrightarrow{OQ} + n\overrightarrow{OP}}{m+n}$.
Substituting the values: $\overrightarrow{OR} = \frac{1(\vec{a} - 2\vec{b}) + 2(2\vec{a} + \vec{b})}{1+2} = \frac{\vec{a} - 2\vec{b} + 4\vec{a} + 2\vec{b}}{3} = \frac{5\vec{a}}{3}$.
(ii) The position vector of the point $R$ dividing the line segment joining $P$ and $Q$ externally in the ratio $m:n = 1:2$ is given by the section formula: $\overrightarrow{OR} = \frac{m\overrightarrow{OQ} - n\overrightarrow{OP}}{m-n}$.
Substituting the values: $\overrightarrow{OR} = \frac{1(\vec{a} - 2\vec{b}) - 2(2\vec{a} + \vec{b})}{1-2} = \frac{\vec{a} - 2\vec{b} - 4\vec{a} - 2\vec{b}}{-1} = \frac{-3\vec{a} - 4\vec{b}}{-1} = 3\vec{a} + 4\vec{b}$.