If ABCD is a quadrilateral, then `vec(BA) + vec(BC)+vec(CD) + vec(DA)=`

Options

A. `2 vec(BA)`
B. `2 vec(AB)`
C. `2vec(AC)`
D. `2(BC)`

Answer

To solve the problem, we need to find the expression for the sum of the vectors in the quadrilateral ABCD, specifically:

\[ \vec{BA} + \vec{BC} + \vec{CD} + \vec{DA} \]

### Step-by-Step Solution:

1. **Understanding the Vectors**:
- We start by recognizing the vectors involved:
- \(\vec{BA} = \vec{A} - \vec{B}\)
- \(\vec{BC} = \vec{C} - \vec{B}\)
- \(\vec{CD} = \vec{D} - \vec{C}\)
- \(\vec{DA} = \vec{A} - \vec{D}\)

2. **Substituting the Vectors**:
- Substitute the expressions for each vector into the equation:
\[ \vec{BA} + \vec{BC} + \vec{CD} + \vec{DA} = (\vec{A} - \vec{B}) + (\vec{C} - \vec{B}) + (\vec{D} - \vec{C}) + (\vec{A} - \vec{D}) \]

3. **Rearranging the Terms**:
- Now, we can rearrange the terms:
\[ = \vec{A} - \vec{B} + \vec{C} - \vec{B} + \vec{D} - \vec{C} + \vec{A} - \vec{D} \]

4. **Combining Like Terms**:
- Combine the like terms:
\[ = \vec{A} + \vec{A} - \vec{B} - \vec{B} + \vec{C} - \vec{C} + \vec{D} - \vec{D} \]
- This simplifies to:
\[ = 2\vec{A} - 2\vec{B} \]

5. **Final Expression**:
- We can factor out the common term:
\[ = 2(\vec{A} - \vec{B}) = 2\vec{BA} \]

### Conclusion:
Thus, the final result is:
\[ \vec{BA} + \vec{BC} + \vec{CD} + \vec{DA} = 2\vec{BA} \]