If $\overrightarrow {\rm{a}}$ and $\overrightarrow {\rm{b}}$ are the position vectors of $\overrightarrow {\rm{A}}$ and $\overrightarrow {\rm{B}}$ respectively, then find the position vector of a point $\overrightarrow {\rm{C}}$ in $\overrightarrow {{\rm{BA}}}$ produced such that $\overrightarrow {{\rm{BC}}} = 1.5\overrightarrow {{\rm{BA}}}$.

Since, $\overrightarrow {{\rm{OA}}} = \overrightarrow {\rm{a}}$ and $\overrightarrow {{\rm{OB}}} = \overrightarrow {\rm{b}}$

$\therefore$ $\overrightarrow {{\rm{BA}}} = \overrightarrow {{\rm{OA}}} - \overrightarrow {{\rm{OB}}} = \overrightarrow {\rm{a}} - \overrightarrow {\rm{b}}$

and $1.5\overrightarrow {{\rm{BA}}} = 1.5(\overrightarrow {\rm{a}} - \overrightarrow {\rm{b}} )$

Since, $\overrightarrow {{\rm{BC}}} = 1.5\overrightarrow {{\rm{BA}}} = 1.5(\overrightarrow {\rm{a}} - \overrightarrow {\rm{b}} )$

$\overrightarrow {{\rm{OC}}} - \overrightarrow {{\rm{OB}}} = 1.5\overrightarrow {\rm{a}} - 1.5\overrightarrow {\rm{b}}$

$\overrightarrow {{\rm{OC}}} = 1.5\overrightarrow {\rm{a}} - 1.5\overrightarrow {\rm{b}} + \overrightarrow {\rm{b}}$

$= 1.5\vec a - 0.5\vec b$
$= \frac{{3\overrightarrow {\rm{a}} - \overrightarrow {\rm{b}} }}{2}$