If a, b, c are the position vectors of the point A, B ,C respectively such that 5a+7b-12c= 0, find the ratio in which the point A divides BC.

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$$\begin{gathered} \mathrm{We} \mathrm{have} \\ \overrightarrow{\mathrm{OA}}=\overrightarrow{\mathrm{a}} \\ \overrightarrow{\mathrm{OB}}=\overrightarrow{\mathrm{b}} \\ \overrightarrow{\mathrm{OC}}=\overrightarrow{\mathrm{c}} \\ \mathrm{Let} \mathrm{ratio} \mathrm{in} \mathrm{which} \mathrm{A} \mathrm{divides} \mathrm{BC} \mathrm{be} \mathrm{\lambda }:1 \\ \overrightarrow{\mathrm{a}}=\frac{\mathrm{\lambda }\overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{b}}}{\mathrm{\lambda }+1} \\ \Rightarrow \overrightarrow{\mathrm{a}}=\frac{\mathrm{\lambda }\overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{b}}}{\mathrm{\lambda }+1} \\ \Rightarrow \overrightarrow{\mathrm{a}}\left(\mathrm{\lambda }+1\right)=\mathrm{\lambda }\overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{b}} \\ \Rightarrow \overrightarrow{\mathrm{a}}\left(\mathrm{\lambda }+1\right)-\overrightarrow{\mathrm{b}}-\mathrm{\lambda }\overrightarrow{\mathrm{c}}=0...\left(\mathrm{i}\right) \\ \mathrm{Given} \mathrm{that} \\ 5\overrightarrow{\mathrm{a}}+7\overrightarrow{\mathrm{b}}-12\overrightarrow{\mathrm{c}}=0 \\ \mathrm{Divide} \mathrm{by} -7 \\ -\frac{5}{7}\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}+\frac{12}{7}\overrightarrow{\mathrm{c}}=0...\left(\mathrm{ii}\right) \\ \mathrm{On} \mathrm{comparing} \left(\mathrm{i}\right) \mathrm{and} \left(\mathrm{ii}\right) \\ \mathrm{\lambda }=-\frac{12}{7} \\ \mathrm{Hence} \mathrm{required} \mathrm{ratio} \mathrm{is} 12:7 \left[\mathrm{Externally}\right] \end{gathered}$$

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