ABCDEF is a regular hexagon with point O as centre. The value of `vec(AB)+vec(AC)+vec(AD)+vec(AE)+vec(AF)` is

Options

A. `2vec(AO)`
B. `2vec(AO)`
C. `6vec(AO)`
D. 0

Explanation

To solve the problem of finding the value of \( \vec{AB} + \vec{AC} + \vec{AD} + \vec{AE} + \vec{AF} \) for a regular hexagon ABCDEF with center O, we can follow these steps:

### Step 1: Understand the Geometry of the Hexagon
A regular hexagon has equal sides and angles. The center O is equidistant from all vertices (A, B, C, D, E, F).

### Step 2: Define the Vectors
- \( \vec{AB} \) is the vector from point A to point B.
- \( \vec{AC} \) is the vector from point A to point C.
- \( \vec{AD} \) is the vector from point A to point D.
- \( \vec{AE} \) is the vector from point A to point E.
- \( \vec{AF} \) is the vector from point A to point F.

### Step 3: Express Each Vector in Terms of Center O
Since O is the center:
- \( \vec{AC} = \vec{AB} + \vec{AO} \)
- \( \vec{AE} = \vec{AF} + \vec{AO} \)

### Step 4: Substitute and Rearrange
Now we can substitute the expressions for \( \vec{AC} \) and \( \vec{AE} \):
\[ \vec{AB} + \vec{AC} + \vec{AD} + \vec{AE} + \vec{AF} = \vec{AB} + (\vec{AB} + \vec{AO}) + \vec{AD} + (\vec{AF} + \vec{AO}) + \vec{AF} \]

### Step 5: Group Similar Terms
Group the vectors:
\[ = \vec{AB} + \vec{AB} + \vec{AO} + \vec{AD} + \vec{AF} + \vec{AO} + \vec{AF} \]
This simplifies to:
\[ = 2\vec{AB} + \vec{AD} + 2\vec{AF} + 2\vec{AO} \]

### Step 6: Analyze the Vectors
Notice that \( \vec{AD} \) can be expressed in terms of \( \vec{AO} \) as well. Since all vertices are symmetric about the center O, we can express \( \vec{AD} \) as \( 2\vec{AO} \) (because it goes from A to D, which is two vertices away).

### Step 7: Combine the Vectors
Now, substituting \( \vec{AD} \):
\[ = 2\vec{AB} + 2\vec{AO} + 2\vec{AF} + 2\vec{AO} \]
This simplifies to:
\[ = 2\vec{AB} + 2\vec{AF} + 4\vec{AO} \]

### Step 8: Final Summation
Since \( \vec{AB} + \vec{AF} \) is equivalent to \( \vec{AO} \) (as they are symmetric), we can write:
\[ = 2\vec{AO} + 4\vec{AO} = 6\vec{AO} \]

### Conclusion
Thus, the value of \( \vec{AB} + \vec{AC} + \vec{AD} + \vec{AE} + \vec{AF} \) is:
\[ \boxed{6\vec{AO}} \]