ABCD is a parallelogram with AC and BD as diagonals. Then, `A vec C - B vec D = `

Options

A. `4 A vec B `
B. ` 3 A vec B `
C. ` 2 A vec B `
D. `A vec B `

Explanation

To solve the problem, we need to find the expression for \( \vec{AC} - \vec{BD} \) in the context of a parallelogram ABCD.

### Step-by-Step Solution:

1. **Identify the Vectors**:
In a parallelogram, the diagonals bisect each other. We can express the vectors \( \vec{AC} \) and \( \vec{BD} \) in terms of the sides of the parallelogram.
- Let \( \vec{A} \) be the position vector of point A, \( \vec{B} \) for point B, \( \vec{C} \) for point C, and \( \vec{D} \) for point D.
- The vector \( \vec{AC} \) can be expressed as:
\[ \vec{AC} = \vec{C} - \vec{A} \]
- The vector \( \vec{BD} \) can be expressed as:
\[ \vec{BD} = \vec{D} - \vec{B} \]

2. **Using the Properties of a Parallelogram**:
In a parallelogram, we have:
\[ \vec{C} = \vec{A} + \vec{AB} + \vec{AD} \]
\[ \vec{D} = \vec{B} + \vec{AD} - \vec{AB} \]
However, we can simplify our approach by using the fact that \( \vec{C} = \vec{A} + \vec{AB} + \vec{AD} \) and \( \vec{D} = \vec{A} + \vec{AD} \).

3. **Express \( \vec{AC} - \vec{BD} \)**:
Now substituting the expressions for \( \vec{AC} \) and \( \vec{BD} \):
\[ \vec{AC} - \vec{BD} = (\vec{C} - \vec{A}) - (\vec{D} - \vec{B}) \]
This simplifies to:
\[ \vec{AC} - \vec{BD} = \vec{C} - \vec{A} - \vec{D} + \vec{B} \]

4. **Substituting for \( \vec{C} \) and \( \vec{D} \)**:
We know that \( \vec{C} = \vec{A} + \vec{AB} + \vec{AD} \) and \( \vec{D} = \vec{B} + \vec{AD} \). Thus:
\[ \vec{AC} - \vec{BD} = (\vec{A} + \vec{AB} + \vec{AD} - \vec{A}) - (\vec{B} + \vec{AD} - \vec{B}) \]
This leads to:
\[ \vec{AC} - \vec{BD} = \vec{AB} - \vec{AD} \]

5. **Final Simplification**:
Since \( \vec{AD} = \vec{BC} \) in a parallelogram, we can conclude:
\[ \vec{AC} - \vec{BD} = \vec{AB} + \vec{BC} - \vec{BC} = \vec{AB} \]

Thus, we find that:
\[ \vec{AC} - \vec{BD} = 2\vec{AB} \]

### Final Answer:
\[ \vec{AC} - \vec{BD} = 2\vec{AB} \]