ABC is a triangle . D divides BC in the ratio l:m, G divides AD in the ratio (l +m) :n .find the position vectors of D and G.

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$$\begin{gathered} \mathrm{Let} \mathrm{position} \mathrm{vector} \mathrm{of} \mathrm{B} \mathrm{is}\overline{ \mathrm{b}} \mathrm{and} \mathrm{position} \mathrm{vector} \mathrm{of} \mathrm{C} \mathrm{is} \overline{\mathrm{c}} \\ \mathrm{So} \\ \mathrm{D} \mathrm{divides} \mathrm{BC} \mathrm{in} \mathrm{the} \mathrm{ratio} \mathrm{l}:\mathrm{m} \\ \mathrm{Hence} \mathrm{position} \mathrm{vector} \mathrm{of} \mathrm{D} =\frac{\mathrm{l}\overline{\mathrm{c}}+\mathrm{m}\overline{\mathrm{b}}}{\mathrm{l}+\mathrm{m}} \mathrm{Answer} \\ \mathrm{Now} \mathrm{Let} \mathrm{position} \mathrm{vector} \mathrm{of} \mathrm{A} \mathrm{is} \overline{\mathrm{a}} \mathrm{and} \mathrm{position} \mathrm{vector} \mathrm{of} \mathrm{D} \mathrm{is} \overline{\mathrm{d}} \\ \mathrm{So} \\ \mathrm{G} \mathrm{divides} \mathrm{AD} \mathrm{in} \mathrm{the} \mathrm{ratio} \left(\mathrm{l}+\mathrm{m}\right):\mathrm{n} \\ \mathrm{Hence} \mathrm{position} \mathrm{vector} \mathrm{of} \mathrm{G}=\frac{\left(\mathrm{l}+\mathrm{m}\right)\overline{\mathrm{d}}+\mathrm{n}\overline{\mathrm{a}}}{\mathrm{l}+\mathrm{m}+\mathrm{n}} \mathrm{Answer} \end{gathered}$$

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