ABCD is a parallelogram . If `vec(AB)=vec(a), vec(BC)=vec(b)`, then what `vec(BD)` equal to ?

Options

A. `vec(a)+vec(b) `
B. `vec(a)-vec(b)`
C. `-vec(a)-vec(b)`
D. `-vec(a)+vec(b)`

Explanation

To solve the problem, we need to find the vector \( \vec{BD} \) in the parallelogram ABCD given that \( \vec{AB} = \vec{a} \) and \( \vec{BC} = \vec{b} \).

### Step-by-Step Solution:

1. **Understand the Parallelogram Properties**:
In a parallelogram, opposite sides are equal and parallel. Therefore, we have:
\[ \vec{AD} = \vec{BC} = \vec{b} \]
and
\[ \vec{AB} = \vec{CD} = \vec{a} \]

2. **Express \( \vec{BD} \)**:
We can express the vector \( \vec{BD} \) using the triangle law of vector addition. In triangle \( ABD \), we have:
\[ \vec{AB} + \vec{BD} + \vec{AD} = 0 \]
Rearranging this gives:
\[ \vec{BD} = -\vec{AB} - \vec{AD} \]

3. **Substitute the Known Vectors**:
We know \( \vec{AB} = \vec{a} \) and \( \vec{AD} = \vec{b} \). Substituting these into the equation gives:
\[ \vec{BD} = -\vec{a} - \vec{b} \]

4. **Rearranging the Expression**:
We can rearrange this to express \( \vec{BD} \) in a more standard form:
\[ \vec{BD} = -\vec{a} + (-\vec{b}) = -\vec{a} + \vec{b} \]

5. **Final Answer**:
Thus, the vector \( \vec{BD} \) can be expressed as:
\[ \vec{BD} = -\vec{a} + \vec{b} \]

### Conclusion:
The final expression for \( \vec{BD} \) is:
\[ \vec{BD} = -\vec{a} + \vec{b} \]