ABCD is a quadrilateral whose diagonals are AC and BD. Which one of the following is correct?

Options

A. A. `vec(BA) + vec(CD) = vec(AC) + vec(DB)`
B. B. `vec(BA) + vec(CD) = vec(BD) + vec(CA)`
C. C. `vec(BA) + vec(CD) = vec(AC) + vec(BD)`
D. D. `vec(BA) + vec(CD) = vec(BC) + vec(AD)`

Answer

To solve the problem, we need to analyze the given relations involving the position vectors of the vertices of the quadrilateral ABCD. The diagonals are AC and BD. We will check the validity of the relations one by one.

### Step-by-Step Solution:

1. **Identify the Position Vectors**:
Let the position vectors of points A, B, C, and D be represented as \( \vec{A}, \vec{B}, \vec{C}, \vec{D} \).

2. **Write the Relations**:
We need to verify the following relation:
\[ \vec{BA} + \vec{CD} = \vec{AC} + \vec{DB} \]

3. **Express the Vectors in Terms of Position Vectors**:
- The vector \( \vec{BA} \) can be expressed as:
\[ \vec{BA} = \vec{A} - \vec{B} \]
- The vector \( \vec{CD} \) can be expressed as:
\[ \vec{CD} = \vec{D} - \vec{C} \]
- The vector \( \vec{AC} \) can be expressed as:
\[ \vec{AC} = \vec{C} - \vec{A} \]
- The vector \( \vec{DB} \) can be expressed as:
\[ \vec{DB} = \vec{B} - \vec{D} \]

4. **Substitute into the Relation**:
Substitute the expressions into the left-hand side (LHS) and right-hand side (RHS):
- LHS:
\[ \vec{BA} + \vec{CD} = (\vec{A} - \vec{B}) + (\vec{D} - \vec{C}) = \vec{A} - \vec{B} + \vec{D} - \vec{C} \]
- Rearranging gives:
\[ \vec{A} - \vec{C} + \vec{D} - \vec{B} \]

5. **Compare with the RHS**:
- RHS:
\[ \vec{AC} + \vec{DB} = (\vec{C} - \vec{A}) + (\vec{B} - \vec{D}) = \vec{C} - \vec{A} + \vec{B} - \vec{D} \]
- Rearranging gives:
\[ -(\vec{A} - \vec{C}) + -(\vec{D} - \vec{B}) \]

6. **Check for Equality**:
- From the LHS, we have:
\[ \vec{A} - \vec{C} + \vec{D} - \vec{B} \]
- From the RHS, we have:
\[ -(\vec{A} - \vec{C}) + -(\vec{D} - \vec{B}) \]
- It can be seen that these two expressions are not equal.

7. **Verify the Second Relation**:
The second relation to check is:
\[ \vec{BA} + \vec{CD} = \vec{BD} + \vec{CA} \]
- LHS:
\[ \vec{BA} + \vec{CD} = (\vec{A} - \vec{B}) + (\vec{D} - \vec{C}) = \vec{A} - \vec{B} + \vec{D} - \vec{C} \]
- RHS:
\[ \vec{BD} + \vec{CA} = (\vec{D} - \vec{B}) + (\vec{A} - \vec{C}) = \vec{D} - \vec{B} + \vec{A} - \vec{C} \]
- Rearranging gives:
\[ \vec{A} - \vec{C} + \vec{D} - \vec{B} \]
- Both LHS and RHS are equal, confirming that the second relation is true.

8. **Conclusion**:
Since the second relation is true and the first is false, we conclude that the correct relation is:
\[ \vec{BA} + \vec{CD} = \vec{BD} + \vec{CA} \]

### Final Answer:
The correct relation is:
\[ \vec{BA} + \vec{CD} = \vec{BD} + \vec{CA} \]