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 vector expression given for the pentagon ABCDE. The expression is:

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

We will simplify this expression step by step.

### Step 1: Rearranging the Vectors
We can rearrange the terms in the expression to group them in a more manageable way. Notice that we can express some vectors in terms of others.

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

### Step 2: Using Vector Properties
We can use the property of vectors that states that the sum of vectors around a closed polygon is zero. In this case, we will consider the path around the pentagon.

1. Start from point A to B: \(\vec{AB}\)
2. Then from B to C: \(\vec{BC}\)
3. From C to A: \(\vec{AC}\)
4. From A to E: \(\vec{AE}\)
5. From E to D: \(\vec{ED}\)
6. Finally from D to C: \(\vec{DC}\)

### Step 3: Grouping the Vectors
Now, we can group the vectors that form a closed loop:

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

### Step 4: Applying the Closure Property
From the properties of vectors in a closed shape, we know that:

\[ \vec{AB} + \vec{BC} + \vec{CD} + \vec{DE} + \vec{EA} = \vec{0} \]

Thus, we can express \(\vec{DC}\) in terms of the other vectors:

\[ \vec{DC} = -(\vec{AB} + \vec{BC} + \vec{AE} + \vec{ED}) \]

### Step 5: Final Simplification
Now we can substitute \(\vec{DC}\) back into our expression:

\[ \vec{AB} + \vec{AE} + \vec{BC} + (-(\vec{AB} + \vec{BC} + \vec{AE} + \vec{ED})) + \vec{ED} + \vec{AC} \]

This simplifies to:

\[ \vec{AC} \]

### Conclusion
Thus, the final result of the expression is:

\[ \vec{AB} + \vec{AE} + \vec{BC} + \vec{DC} + \vec{ED} + \vec{AC} = 3\vec{AC} \]