\[\begin{array}{l}
a)\overrightarrow {AB} .\overrightarrow {AD}  = \left| {\overrightarrow {AB} } \right|.\left| {\overrightarrow {AD} } \right|.\cos \left( {\overrightarrow {AB} ,\overrightarrow {AD} } \right)\\
 = a.a.\cos {120^0} = {a^2}.\left( { - \dfrac{1}{2}} \right) =  - \dfrac{{{a^2}}}{2}\\
\overrightarrow {AM}  = \dfrac{1}{2}\left( {\overrightarrow {AB}  + \overrightarrow {AC} } \right) = \dfrac{1}{2}\overrightarrow {AB}  + \left( {\overrightarrow {AB}  + \overrightarrow {AD} } \right) = \dfrac{3}{2}\overrightarrow {AB}  + \overrightarrow {AD} \\
\overrightarrow {AN}  = \overrightarrow {AD}  + \overrightarrow {DN}  = \overrightarrow {AD}  + \dfrac{1}{3}\overrightarrow {DC}  = \overrightarrow {AD}  + \dfrac{1}{3}\overrightarrow {AB} \\
 \Rightarrow \overrightarrow {AM} .\overrightarrow {AN}  = \left( {\dfrac{3}{2}\overrightarrow {AB}  + \overrightarrow {AD} } \right)\left( {\overrightarrow {AD}  + \dfrac{1}{3}\overrightarrow {AB} } \right)\\
 = \dfrac{3}{2}\overrightarrow {AB} .\overrightarrow {AD}  + A{D^2} + \dfrac{1}{2}A{B^2} + \overrightarrow {AD} .\overrightarrow {AB} \\
 = \dfrac{5}{2}\overrightarrow {AB} .\overrightarrow {AD}  + A{D^2} + \dfrac{1}{2}A{B^2}\\
 = \dfrac{5}{2}.\left( { - \dfrac{{{a^2}}}{2}} \right) + {a^2} + \dfrac{1}{2}{a^2} = \dfrac{{{a^2}}}{4}\\
b)I\left( {0;y} \right) \in Oy\\
\overrightarrow {AI}  = \left( {3;y + 1} \right),\overrightarrow {BI}  = \left( {1;y + 5} \right)\\
AI \bot BI \Leftrightarrow \overrightarrow {AI} .\overrightarrow {BI}  = 0 \Leftrightarrow 3 + \left( {y + 1} \right)\left( {y + 5} \right) = 0\\
 \Leftrightarrow {y^2} + 6y + 8 = 0 \Leftrightarrow \left[ \begin{array}{l}
y =  - 2\\
y =  - 4
\end{array} \right. \Rightarrow \left[ \begin{array}{l}
I\left( {0; - 2} \right)\\
I\left( {0; - 5} \right)
\end{array} \right.
\end{array}\]