To solve for $\vec{AM}$ in terms of $a$ and $b$:

1. We know that $M$ is the midpoint of $AC$, which means: $\vec{AM} = \frac{1}{2} \vec{AC}$

2. To find $\vec{AC}$, we need to add $\vec{AB}$ and $\vec{BC}$ because $A \to B \to C$ forms a path from $A$ to $C$: $\vec{AC} = \vec{AB} + \vec{BC}$

3. Substitute the given values for $\vec{AB}$ and $\vec{BC}$: $\vec{AC} = (8a - 6b) + 20b$

4. Combine like terms: $\vec{AC} = 8a + (20b - 6b) = 8a + 14b$

5. Now, find $\vec{AM}$ by taking half of $\vec{AC}$: $\vec{AM} = \frac{1}{2}(8a + 14b)$

6. Simplify: $\vec{AM} = 4a + 7b$

Final Answer:

$\vec{AM} = 4a + 7b$

Would you like more details on any of these steps, or do you have further questions?

Here are five related questions to expand your understanding:

1. How would the solution change if $M$ was not the midpoint of $AC$?
2. Can you apply similar vector addition concepts to find the midpoint of any segment in a polygon?
3. What if $AB$ and $BC$ were in terms of different variables—how would that affect the simplification?
4. How would this approach work if you only had coordinates for points $A$, $B$, and $C$?
5. What does it mean geometrically when we take half of $\vec{AC}$ to find $\vec{AM}$?

Tip: When dealing with midpoints in vector problems, you can always simplify by taking half of the entire length of the segment rather than adding individual vector components.