a, Gọi D là trung điểm của MN \(\Rightarrow\overrightarrow{MN}=2\overrightarrow{MD}\).

Ta có: \(\overrightarrow{NA}+3\overrightarrow{NC}=\overrightarrow{0}\Leftrightarrow\overrightarrow{AN}=3\overrightarrow{NC}\) \(\Leftrightarrow AN=3NC\)

\(\overrightarrow{MD}=\overrightarrow{AD}-\overrightarrow{AM}=\frac{1}{2}\left(\overrightarrow{AM}+\overrightarrow{AN}\right)-\overrightarrow{AM}=\frac{1}{2}\overrightarrow{AN}-\frac{1}{2}\overrightarrow{AM}\)

\(\overrightarrow{MD}=\frac{3}{8}AC-\frac{1}{4}\overrightarrow{AB}\Rightarrow\overrightarrow{MN}=\frac{3}{4}\overrightarrow{AC}-\frac{1}{2}\overrightarrow{AB}\)

b, IM là đường trung bình của tam giác ABC

\(\Rightarrow\) \(\overrightarrow{IM}=\frac{1}{2}\overrightarrow{CA}=\frac{1}{2}\left(\overrightarrow{IA}-\overrightarrow{IC}\right)\)

Tham khảo:

Dễ thấy: \(\overrightarrow {OA}  = \overrightarrow {OM}  + \overrightarrow {MA} \); \(\overrightarrow {OB}  = \overrightarrow {OM}  + \overrightarrow {MB} \)

Tương tự: \(\overrightarrow {OC}  = \overrightarrow {ON}  + \overrightarrow {NC} \); \(\overrightarrow {OD}  = \overrightarrow {ON}  + \overrightarrow {ND} \)

\(\begin{array}{l} \Rightarrow \overrightarrow {OA}  + \overrightarrow {OB}  + \overrightarrow {OC}  + \overrightarrow {OD}  = \left( {\overrightarrow {OM}  + \overrightarrow {MA} } \right) + \left( {\overrightarrow {OM}  + \overrightarrow {MB} } \right) + \left( {\overrightarrow {ON}  + \overrightarrow {NC} } \right) + \left( {\overrightarrow {ON}  + \overrightarrow {ND} } \right)\\ = \left( {\overrightarrow {OM}  + \overrightarrow {OM}  + \overrightarrow {MA}  + \overrightarrow {MB} } \right) + \left( {\overrightarrow {ON}  + \overrightarrow {ON}  + \overrightarrow {NC}  + \overrightarrow {ND} } \right)\\ = \overrightarrow {OM}  + \overrightarrow {OM}  + \overrightarrow {ON}  + \overrightarrow {ON} \\ = \left( {\overrightarrow {OM}  + \overrightarrow {ON} } \right) + \left( {\overrightarrow {OM}  + \overrightarrow {ON} } \right)\\ = \overrightarrow 0  + \overrightarrow 0 \\ = \overrightarrow 0 .\end{array}\)