question_answer

If \[\vec{a}\] and \[\vec{b}\] are the position vectors of A and B respectively, find the position vector of a point C on BA produced such that BC = 1.5 BA

Answer:

Let \[\overrightarrow{OA}=\vec{a},\,\,\overrightarrow{OB}=\vec{b}\] and \[\overrightarrow{OC}=\vec{c}\] We have, \[\overrightarrow{BC}=1.5\,\,\overrightarrow{BA}\] \[\because \]       \[\overrightarrow{BC}=\overrightarrow{OC}-\overrightarrow{OB}\] and       \[\overrightarrow{BA}=\overrightarrow{OA}-\overrightarrow{OB}\] \[\therefore \]      \[\overrightarrow{OC}-\overrightarrow{OB}=1.5(\overrightarrow{OA}-\overrightarrow{OB})\] \[\Rightarrow \]   \[\overrightarrow{OC}-\vec{b}=1.5(\vec{a}-\vec{b})\] \[\Rightarrow \]   \[\overrightarrow{OC}=1.5\vec{a}\,\,-1.5\,\vec{b}+\vec{b}\]             \[=1.5\vec{a}\,\,-0.5\,\vec{b}\]             \[=\frac{3}{2}\vec{a}\,\,-\frac{1}{2}\,\vec{b}\] \[\Rightarrow \]   \[\vec{c}=\frac{3\vec{a}-\vec{b}}{2}\]