Question

The position vector of the point which divides the join of points given by position vectors $2\overrightarrow{a}-3\overrightarrow{b}$ and $3\overrightarrow{a}-2\overrightarrow{b}$ internally in the ratio 2:3 is

• A. $\frac{-13}{5}\overrightarrow{a}+\frac{12}{5}\overrightarrow{b}$
• B. $\frac{12}{5}\overrightarrow{a}-\frac{13}{5}\overrightarrow{b}$
• C. $\frac{12}{13}\overrightarrow{a}-\frac{13}{12}\overrightarrow{b}$
• D. None of the above

Answer 1

The correct option is B $\frac{12}{5}\overrightarrow{a}-\frac{13}{5}\overrightarrow{b}$

Using the section formula, the position vector of the point which divides the line segment joining the points given by the position vectors $2\overrightarrow{a}-3\overrightarrow{b}$ and $3\overrightarrow{a}-2\overrightarrow{b}$

In the ratio 2:3 will be given by
$\frac{3\overrightarrow{A}+2\overrightarrow{B}}{2+3}$
=$\frac{3(2\overrightarrow{a}-3\overrightarrow{b})+2(3\overrightarrow{a}-2\overrightarrow{b})}{2+3}$
=$\frac{6\overrightarrow{a}+6\overrightarrow{a}-9\overrightarrow{b}-4\overrightarrow{b}}{5}$
=$\frac{12}{5}\overrightarrow{a}-\frac{13}{5}\overrightarrow{b}$