In a quadrilateral ABCD, `vec(AB) + vec(DC) =`

Options

A. `vec(AB) + vec(CB)`
B. `vec(AC) + vec(BD)`
C. `vec(AC) + vec(DB)`
D. `vec(AD) - vec(CB)`

Answer

To solve the problem, we need to express the vector sum \( \vec{AB} + \vec{DC} \) in terms of other vectors in the quadrilateral ABCD.

### Step-by-Step Solution:

1. **Identify the vectors**:
We have the vectors \( \vec{AB} \), \( \vec{BC} \), \( \vec{CD} \), and \( \vec{DA} \) in the quadrilateral ABCD.

2. **Express \( \vec{DC} \)**:
The vector \( \vec{DC} \) can be expressed in terms of \( \vec{CD} \):
\[ \vec{DC} = -\vec{CD} \]

3. **Combine the vectors**:
Now we can write the expression for \( \vec{AB} + \vec{DC} \):
\[ \vec{AB} + \vec{DC} = \vec{AB} - \vec{CD} \]

4. **Add and subtract \( \vec{BC} \)**:
We can add and subtract \( \vec{BC} \) to the equation:
\[ \vec{AB} + \vec{DC} = \vec{AB} + \vec{BC} - \vec{BC} - \vec{CD} \]

5. **Group the vectors**:
Rearranging gives us:
\[ = (\vec{AB} + \vec{BC}) - (\vec{BC} + \vec{CD}) \]

6. **Use triangle law of vector addition**:
From the triangle law of vector addition:
\[ \vec{AB} + \vec{BC} = \vec{AC} \]
and
\[ \vec{BC} + \vec{CD} = \vec{BD} \]
Therefore, we can substitute:
\[ = \vec{AC} - \vec{BD} \]

7. **Express \( \vec{BD} \)**:
Since \( \vec{BD} = -\vec{DB} \), we can rewrite the equation:
\[ = \vec{AC} + \vec{DB} \]

8. **Final result**:
Thus, we conclude that:
\[ \vec{AB} + \vec{DC} = \vec{AC} + \vec{DB} \]

### Final Answer:
\[ \vec{AB} + \vec{DC} = \vec{AC} + \vec{DB} \]