If `veca` lies in the plane of vectors `vecb` and `vecc`, then which of the following is correct?

Options

A. `[(veca,vecb, vecc)]=0`
B. `[(veca, vecb, vecc)]=1`
C. `[(veca, vecb, vecc)]=3`
D. `[(veca, vecc, veca)]=1`

Explanation

To solve the problem, we need to analyze the relationship between the vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) given that \(\vec{a}\) lies in the plane formed by \(\vec{b}\) and \(\vec{c}\).

### Step-by-Step Solution:

1. **Understanding Coplanarity**:
- Since \(\vec{a}\) lies in the plane of \(\vec{b}\) and \(\vec{c}\), it implies that the vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) are coplanar.

2. **Cross Product**:
- The cross product \(\vec{b} \times \vec{c}\) produces a vector that is perpendicular to the plane formed by \(\vec{b}\) and \(\vec{c}\).

3. **Perpendicularity**:
- Since \(\vec{a}\) is in the same plane as \(\vec{b}\) and \(\vec{c}\), it follows that the vector \(\vec{b} \times \vec{c}\) must be perpendicular to \(\vec{a}\).

4. **Dot Product**:
- If two vectors are perpendicular, their dot product is zero. Therefore, we can write:
\[ \vec{a} \cdot (\vec{b} \times \vec{c}) = 0 \]

5. **Scalar Triple Product**:
- The expression \(\vec{a} \cdot (\vec{b} \times \vec{c})\) is known as the scalar triple product of the vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\).
- Since we established that \(\vec{a} \cdot (\vec{b} \times \vec{c}) = 0\), we conclude that:
\[ [\vec{a}, \vec{b}, \vec{c}] = 0 \]
- This indicates that the scalar triple product of \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) is zero.

6. **Conclusion**:
- Therefore, the correct statement regarding the vectors is that the scalar triple product \([\vec{a}, \vec{b}, \vec{c}] = 0\).

### Final Answer:
The correct option is that the scalar triple product \([\vec{a}, \vec{b}, \vec{c}] = 0\).

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