If ABCDE is a pentagon, then
`vec(AB) + vec(AE) + vec(BC) + vec(DC) + vec(ED) + vec(AC) ` is equal to

Options

A. `4 vec(AC)`
B. `2 vec(AC)`
C. `3vec(AC)`
D. `5 vec(AC)`

Answer

To solve the problem, we need to analyze the vectors given in the pentagon ABCDE and simplify the expression:

\[ \vec{AB} + \vec{AE} + \vec{BC} + \vec{DC} + \vec{ED} + \vec{AC} \]

### Step 1: Understand the vectors in the pentagon
The pentagon ABCDE consists of the following sides:
- \(\vec{AB}\) from A to B
- \(\vec{AE}\) from A to E
- \(\vec{BC}\) from B to C
- \(\vec{DC}\) from D to C
- \(\vec{ED}\) from E to D
- \(\vec{AC}\) from A to C

### Step 2: Rearranging the vectors
We can rearrange the vectors in a way that groups them logically. We can express the vectors in terms of the segments that connect the points A, B, C, D, and E.

\[ \vec{AB} + \vec{AE} + \vec{BC} + \vec{DC} + \vec{ED} + \vec{AC} = \vec{AB} + \vec{AE} + \vec{AC} + \vec{BC} + \vec{DC} + \vec{ED} \]

### Step 3: Combine vectors
Notice that we can combine some vectors based on their connections:
- \(\vec{AB} + \vec{BC} = \vec{AC}\)
- \(\vec{AE} + \vec{ED} + \vec{DC}\) can also be rearranged to form paths that connect back to point A.

### Step 4: Form triangles
We can visualize the arrangement of these vectors. The vectors \(\vec{AB}\) and \(\vec{AC}\) can be seen as forming a triangle with point B. Similarly, \(\vec{AE}\), \(\vec{ED}\), and \(\vec{DC}\) can form another triangle with point D.

### Step 5: Final simplification
From the rearrangement and combination of vectors, we can conclude that:
\[ \vec{AB} + \vec{AE} + \vec{BC} + \vec{DC} + \vec{ED} + \vec{AC} = 3 \vec{AC} \]

### Conclusion
Thus, the expression simplifies to:
\[ \vec{AB} + \vec{AE} + \vec{BC} + \vec{DC} + \vec{ED} + \vec{AC} = 3 \vec{AC} \]