If $\vec{a}, \vec{b}, \vec{c}, \vec{d}$ are position vectors of 4 points such that $2 \vec{a}+3 \vec{b}+5…
If $\vec{a}, \vec{b}, \vec{c}, \vec{d}$ are position vectors of 4 points such that $2 \vec{a}+3 \vec{b}+5 \vec{c}-10 \vec{d}=\overrightarrow{0}$ then the ratio in which the line joining $\vec{c}$ and $\vec{d}$ divides the line segment joining $\vec{a}$ and $\vec{b}$ is
$2: 3$
$-1: 2$
$2: 1$
$3: 2$
Solution
$\begin{aligned}
& 2 \vec{a}+3 \vec{b}+5 \vec{c}-10 \vec{d}=\overrightarrow{0}, \Rightarrow 2 \vec{a}+3 \vec{b}=10 \vec{d}-5 \vec{c} \\
& \Rightarrow \frac{2 \vec{a}+3 \vec{b}}{5}=\frac{2 \vec{d}-\vec{c}}{1} \Rightarrow \frac{2 \vec{a}+3 \vec{b}}{3+2}=\frac{2 \vec{d}-\vec{c}}{2-1}
\end{aligned}$
$\therefore$ Line joining $\vec{c}$ and $\vec{d}$ divides the line joining $\vec{a}$ and $\vec{b}$ in 3:2.