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

"$BA$ produced" means extend the segment $BA$ beyond $A$, keeping the direction from $B$ to $A$. So $\vec{BC}=1.5\,\vec{BA}$.

$\vec{BA}=\vec{a}-\vec{b}$, hence $\vec{BC}=1.5(\vec{a}-\vec{b})$.

Position vector of $C$:

$\vec{c}=\vec{b}+\vec{BC}=\vec{b}+1.5(\vec{a}-\vec{b})=1.5\vec{a}-0.5\vec{b}=\frac{3\vec{a}-\vec{b}}{2}$ …

Reading the problem

$A$ and $B$ have position vectors $\vec{a}$ and $\vec{b}$. "$BA$ produced" means we travel from $B$ towards $A$ and keep going past $A$. So $C$ lies on that ray, in the same direction as $\vec{BA}$, at a distance $BC=1.5\,BA$.

Set up the displacement

The step from $B$ to $A$ is

Because $C$ is along this same direction with $BC=1.5\,BA$,

Find the position vector of C

Sanity check …

Method: Finding a point on a produced line by adding a scaled displacement

Use this when a point is described as lying on a line "produced" a given multiple of a segment — e.g. $C$ on $BA$ produced with $BC=1.5\,BA$.

Steps

Step 1: Decode the wording into a direction.

"$BA$ produced" means travel from $B$ toward $A$ and continue past $A$; the direction of motion is $\vec{BA}=\vec{a}-\vec{b}$ (position of $A$ minus position of $B$). Getting this order right fixes the sign of everything that follows.

Step 2: Scale the base displacement by the given ratio. …

Common Mistakes

Mistake 1: Taking the direction as $\vec{AB}=\vec{b}-\vec{a}$ instead of $\vec{BA}=\vec{a}-\vec{b}$.

Why it's wrong: "$BA$ produced" travels from $B$ toward $A$, so the direction is $\vec{a}-\vec{b}$; the opposite sign places $C$ on the wrong side. Correct approach: read the segment order $BA$ and set $\vec{BC}=1.5(\vec{a}-\vec{b})$.

Mistake 2: Placing $B$ between $A$ and $C$. …