Question

OAB is a given triangle such that $\overrightarrow{OA}=\overline{a}$, $\overrightarrow{OB}=\overline{b}$.
Also C is a point on AB such that $\overrightarrow{AB}=2\overrightarrow{BC}$. State which of the following statements are correct?

• A. $\overrightarrow{AC}=\frac{2}{3}(\overline{b}-\overline{a})$,
• B. $\overrightarrow{AC}=\frac{2}{3}(\overline{a}-\overline{b})$,
• C. $\overrightarrow{AC}=\frac{2}{3}(\overline{b}+\overline{a})$,
• D. $\overrightarrow{AC}=\frac{3}{2}(\overline{b}-\overline{a})$,

Answer 1

Answer: (D)

$\overrightarrow{AC}=\frac{3}{2}(\overline{b}-\overline{a})$,

Given $\overline{OA}=\overline{a}$ and $\overline{OB}=\overline{b}$
$C$ is apoint on $AB$ such that $\overline{AB}=2\overline{BC}$
$\Rightarrow \overline{b}-\overline{a}=2(\overline{c}-\overline{b})$
$\Rightarrow \overline{c}=\frac{3\overline{b}-\overline{a}}{2}$
$\therefore \overline{AC}=\overline{c}-\overline{a}=\frac{3}{2}(\overline{b}-\overline{a})$
Hence, option D.