If `a .(b xx c) = 0`, then the correct statement is

Options

A. out of a, b, c, any two vectors are parallel
B. a, b, c are coplanar
C. any two are equal among a,b,c
D. atleast one statement is correct

Answer

To solve the problem, we need to analyze the given condition \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = 0 \) and determine the implications of this condition regarding the vectors involved.

### Step-by-Step Solution:

1. **Understanding the Scalar Triple Product**:
The expression \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \) represents the scalar triple product of the vectors \( \mathbf{a}, \mathbf{b}, \) and \( \mathbf{c} \).

2. **Condition for Coplanarity**:
The scalar triple product \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = 0 \) indicates that the volume of the parallelepiped formed by the vectors \( \mathbf{a}, \mathbf{b}, \) and \( \mathbf{c} \) is zero. This occurs if and only if the vectors are coplanar.

3. **Conclusion**:
Since the scalar triple product is zero, we conclude that the vectors \( \mathbf{a}, \mathbf{b}, \) and \( \mathbf{c} \) are coplanar.

4. **Evaluating Options**:
- Option 1: Any two vectors are parallel. (Not necessarily true)
- Option 2: \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) are coplanar. (True)
- Option 3: Any two are equal among \( \mathbf{a}, \mathbf{b}, \mathbf{c} \). (Not necessarily true)
- Option 4: None of the above. (False since option 2 is true)

Thus, the correct statement is that \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) are coplanar.

### Final Answer:
The correct statement is: **\( \mathbf{a}, \mathbf{b}, \mathbf{c} \) are coplanar.**

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