If $\vec{a}$ and $\vec{b}$ are the position vectors of $A$ and $B$ respectively, then find the position vector of a point $C$ in $BA$ produced such that $BC = 1.5 BA$.
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → $\frac{3\vec{a} - \vec{b}}{2}$ ##
Since, $\vec{OA} = \vec{a}$ and $\vec{OB} = \vec{b}$
$∴\vec{BA} = \vec{OA} - \vec{OB} = \vec{a} - \vec{b}$
And since, $1.5 \vec{BA} = 1.5(\vec{a} - \vec{b})$ [multiply by 1.5 on both sides of equation]
$\vec{BC} = 1.5 \vec{BA} = 1.5(\vec{a} - \vec{b})$
$\vec{OC} - \vec{OB} = 1.5\vec{a} - 1.5\vec{b}$
$\vec{OC} = 1.5\vec{a} - 1.5\vec{b} + \vec{b}$ $[∵\vec{OB} = \vec{b}]$
$= 1.5\vec{a} - 0.5\vec{b}$
$= \frac{3\vec{a} - \vec{b}}{2}$
