$$\eqalign{
& \overrightarrow {IA} + 3\overrightarrow {IB} = \overrightarrow 0 \cr
& \Leftrightarrow 3\overrightarrow {IB} = \overrightarrow {AI} \Leftrightarrow \overrightarrow {IB} = {1 \over 3}\overrightarrow {AI} \cr
& a)\,\,\overrightarrow {AI} = k\overrightarrow {AB} \cr
& \Leftrightarrow \overrightarrow {AI} = k\left( {\overrightarrow {AI} + \overrightarrow {IB} } \right) \cr
& \Leftrightarrow \overrightarrow {AI} = k\overrightarrow {AI} + k\overrightarrow {IB} \cr
& \Leftrightarrow \overrightarrow {AI} = k\overrightarrow {AI} + {1 \over 3}\overrightarrow {AI} \cr
& \Leftrightarrow \overrightarrow {AI} \left( {1 - k - {1 \over 3}} \right) = 0 \cr
& \Leftrightarrow \overrightarrow {AI} \left( {{2 \over 3} - k} \right) = 0 \cr
& \Leftrightarrow {2 \over 3} - k = 0 \Leftrightarrow k = {2 \over 3} \cr
& b)\,\,\overrightarrow {MI} = {2 \over 5}\overrightarrow {MA} + {3 \over 5}\overrightarrow {MB} \cr
& \Leftrightarrow \overrightarrow {MI} = {2 \over 5}\left( {\overrightarrow {MI} + \overrightarrow {IA} } \right) + {3 \over 5}\left( {\overrightarrow {MI} + \overrightarrow {IB} } \right) \cr
& \Leftrightarrow \overrightarrow {MI} = {2 \over 5}\overrightarrow {MI} + {2 \over 5}\overrightarrow {IA} + {3 \over 5}\overrightarrow {MI} + {3 \over 5}\overrightarrow {IB} \cr
& \Leftrightarrow \overrightarrow {MI} = \overrightarrow {MI} + {1 \over 5}\left( {2\overrightarrow {IA} + 3\overrightarrow {IB} } \right) \cr
& \Leftrightarrow \overrightarrow {MI} = \overrightarrow {MI} \,\,\left( {luon\,\,dung} \right) \cr
& Vay\,\,\,\overrightarrow {MI} = {2 \over 5}\overrightarrow {MA} + {3 \over 5}\overrightarrow {MB} \,\,\forall M \cr} $$
