1. **Problem statement:** Given a prism ABC.A'B'C' with M as the midpoint of BB'. Given vectors $\overrightarrow{AC} = \vec{a}, \overrightarrow{BC} = \vec{b}, \overrightarrow{AA'} = \vec{c}$, find the correct expression for $\overrightarrow{AM}$. 2. **Understanding the problem:** We want to express $\overrightarrow{AM}$ in terms of $\vec{a}, \vec{b}, \vec{c}$. 3. **Step 1: Express points in vector form relative to A.** - $\overrightarrow{AC} = \vec{a}$ means $\vec{c} = \overrightarrow{AA'}$. - $\overrightarrow{BC} = \vec{b}$. 4. **Step 2: Find $\overrightarrow{AB}$ in terms of $\vec{a}, \vec{b}$.** Since $\overrightarrow{BC} = \vec{b}$ and $\overrightarrow{AC} = \vec{a}$, then $$\overrightarrow{AB} = \overrightarrow{AC} - \overrightarrow{BC} = \vec{a} - \vec{b}.$$ 5. **Step 3: Find $\overrightarrow{AM}$.** Point M is midpoint of BB', so $$\overrightarrow{AM} = \overrightarrow{AB} + \frac{1}{2} \overrightarrow{BB'}.$$ 6. **Step 4: Express $\overrightarrow{BB'}$ in terms of $\vec{c}$.** Since $\overrightarrow{AA'} = \vec{c}$ and BB' is parallel and equal to AA', $$\overrightarrow{BB'} = \vec{c}.$$ 7. **Step 5: Substitute into $\overrightarrow{AM}$:** $$\overrightarrow{AM} = (\vec{a} - \vec{b}) + \frac{1}{2} \vec{c} = \vec{a} - \vec{b} + \frac{1}{2} \vec{c}.$$ 8. **Answer:** The correct expression is $\overrightarrow{AM} = \vec{a} - \vec{b} + \frac{1}{2} \vec{c}$, which corresponds to option A.