If the vector `vec(a)` lies in the planar of vectors `vec(b) and vec(c )`, then which one of the following is correct?

Options

A. `vec(a).(vec(b) xx vec(c )) =0`
B. `vec(a).(vec(b) xx vec(c ))=1`
C. `vec(a) (vec(b) xx vec(c ))=-1`
D. `vec(a) (vec(b) xx vec(a)) =3`

Answer

To solve the problem, we need to determine 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**:
- If a vector \( \vec{a} \) lies in the plane of vectors \( \vec{b} \) and \( \vec{c} \), it means that \( \vec{a} \) can be expressed as a linear combination of \( \vec{b} \) and \( \vec{c} \). This implies that the three vectors \( \vec{a} \), \( \vec{b} \), and \( \vec{c} \) are coplanar.

2. **Scalar Triple Product**:
- The scalar triple product of three vectors \( \vec{a} \), \( \vec{b} \), and \( \vec{c} \) is given by the formula:
\[ \vec{a} \cdot (\vec{b} \times \vec{c}) \]
- This scalar triple product represents the volume of the parallelepiped formed by the three vectors.

3. **Condition for Coplanarity**:
- For vectors to be coplanar, the volume of the parallelepiped they form must be zero. Therefore, the scalar triple product must equal zero:
\[ \vec{a} \cdot (\vec{b} \times \vec{c}) = 0 \]

4. **Conclusion**:
- Since \( \vec{a} \) lies in the plane of \( \vec{b} \) and \( \vec{c} \), we conclude that:
\[ \vec{a} \cdot (\vec{b} \times \vec{c}) = 0 \]
- Therefore, the correct answer is that the scalar triple product \( \vec{a} \cdot (\vec{b} \times \vec{c}) \) is zero.

### Final Answer:
The correct option is:
\[ \vec{a} \cdot (\vec{b} \times \vec{c}) = 0 \]