Question

$A$ and $B$ are two points. The position vector of $A$ is $6\overrightarrow{b}-2\overrightarrow{a}$. A point $P$ divides the line $\overrightarrow{AB}$ in the ratio $1:2.$ If $\overrightarrow{a}-\overrightarrow{b}$ is the position vector of $P$, then the position vector of $B$ is given by

• A. $7\overrightarrow{a}-15\overrightarrow{b}$
• B. $7\overrightarrow{a}+15\overrightarrow{b}$
• C. $15\overrightarrow{a}-7\overrightarrow{b}$
• D. $15\overrightarrow{a}+7\overrightarrow{b}$

Answer 1

The correct option is A $7\overrightarrow{a}-15\overrightarrow{b}$

Let $B$ be the point $\left(\overrightarrow{r}\right)$

$\therefore \frac{1\cdot \overrightarrow{r}+2(6\overrightarrow{b}-2\overrightarrow{a})}{1+2}=\overrightarrow{a}-\overrightarrow{b}$
$\Rightarrow \overrightarrow{r}+12\overrightarrow{b}-4\overrightarrow{a}=3\overrightarrow{a}-3\overrightarrow{b}$
$\Rightarrow \overrightarrow{r}=7\overrightarrow{a}-15\overrightarrow{b}$
$\therefore B$ is $7\overrightarrow{a}-15\overrightarrow{b}$