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1.
\(\left\{{}\begin{matrix}\overrightarrow{BA}+\overrightarrow{BC}=2\overrightarrow{BN}\\\overrightarrow{CA}+\overrightarrow{CB}=2\overrightarrow{CP}\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}-\overrightarrow{AB}+\overrightarrow{BC}=2\overrightarrow{BN}\\\overrightarrow{CB}+\overrightarrow{BA}+\overrightarrow{CB}=2\overrightarrow{CP}\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}\overrightarrow{AB}-\overrightarrow{BC}=-2\overrightarrow{BN}\\\overrightarrow{AB}+2\overrightarrow{BC}=-2\overrightarrow{CP}\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}2\overrightarrow{AB}-2\overrightarrow{BC}=-4\overrightarrow{BN}\\\overrightarrow{AB}+2\overrightarrow{BC}=-2\overrightarrow{CP}\end{matrix}\right.\)
\(\Rightarrow3\overrightarrow{AB}=-4\overrightarrow{BN}-2\overrightarrow{CP}\Rightarrow\overrightarrow{AB}=-\frac{4}{3}\overrightarrow{BN}-\frac{2}{3}\overrightarrow{CP}\)
2.
\(\overrightarrow{BI}=\overrightarrow{BA}+\overrightarrow{AD}+\overrightarrow{DI}\)
\(=-\overrightarrow{AB}+\overrightarrow{AD}+\frac{1}{2}\overrightarrow{DC}\)
\(=-\overrightarrow{AB}+\overrightarrow{AD}+\frac{1}{2}\overrightarrow{AB}\)
\(\Rightarrow\overrightarrow{BI}=-\frac{1}{2}\overrightarrow{AB}+\overrightarrow{AD}\)
\(\overrightarrow{AG}=\overrightarrow{AB}+\overrightarrow{BG}=\overrightarrow{AB}+\frac{1}{3}\left(\overrightarrow{BI}+\overrightarrow{BC}\right)\)
\(=\overrightarrow{AB}+\frac{1}{3}\left(-\frac{1}{2}\overrightarrow{AB}+\overrightarrow{AD}+\overrightarrow{AD}\right)\)
\(=\overrightarrow{AB}-\frac{1}{6}\overrightarrow{AB}+\frac{2}{3}\overrightarrow{AD}\)
\(\Rightarrow\overrightarrow{AG}=\frac{5}{6}\overrightarrow{AB}+\frac{2}{3}\overrightarrow{AD}\)