If \(2 \overrightarrow{\mathbf{a}}+3 \overrightarrow{\mathbf{b}}-5 \overrightarrow{\mathbf{c}}=\overrightarrow{\mathbf{0}}\), then ratio in which \(\overrightarrow{\mathbf{c}}\)
divides \(\overrightarrow{\mathbf{A B}}\) is

Given, \(2 \overrightarrow{\mathbf{a}}+3 \overrightarrow{\mathbf{b}}-5 \overrightarrow{\mathbf{c}}=0\)
\(\Rightarrow \frac{2 \overrightarrow{\mathbf{a}}+3 \overrightarrow{\mathbf{b}}}{5}=\overrightarrow{\mathbf{c}}\)
\(\Rightarrow \frac{2 \overrightarrow{\mathbf{a}}+3 \overrightarrow{\mathbf{b}}}{2+3}=\overrightarrow{\mathbf{c}}\)
\(\Rightarrow \quad \frac{\overrightarrow{\mathbf{a}}+\frac{3}{2} \overrightarrow{\mathbf{b}}}{1+\frac{3}{2}}=\overrightarrow{\mathbf{c}}\)


Let \(\overrightarrow{\mathbf{c}}\) divide \(\overrightarrow{\mathbf{A B}}\) in the ratio \(\lambda: 1\)
Then, \(\overrightarrow{\mathbf{c}}=\frac{\overrightarrow{\mathbf{a}}+\lambda \overrightarrow{\mathbf{b}}}{1+\lambda}\)
On comparing Eqs. (i) and (ii), we get \(\lambda=\frac{3}{2}\)
\(\therefore\) Required ratio is \(3: 2\) internally.