If `alpha(axxb)+beta(bxxc)+gamma(cxxa)=0`, then

Options

A. a,b,c are coplanar if all of `alpha,beta,gamma ne0`
B. a,b,c are coplanar if any one of `alpha,beta,gammane0`
C. a,b,c are non-coplanar for any `alpha,beta,gammane0`
D. none of these

Answer

To solve the problem given by the equation \( \alpha ( \mathbf{a} \times \mathbf{b}) + \beta ( \mathbf{b} \times \mathbf{c}) + \gamma ( \mathbf{c} \times \mathbf{a}) = 0 \), we need to analyze the implications of this vector equation.

### Step-by-step Solution:

1. **Understanding the Vector Equation**:
The equation states that a linear combination of the cross products of vectors \( \mathbf{a}, \mathbf{b}, \) and \( \mathbf{c} \) equals the zero vector. This implies that the vectors are linearly dependent.

**Hint**: Recall that if a linear combination of vectors equals zero, the vectors must be coplanar.

2. **Using the Scalar Triple Product**:
We can express the cross products in terms of the scalar triple product. The scalar triple product \( [\mathbf{a}, \mathbf{b}, \mathbf{c}] \) (or \( \text{box}( \mathbf{a}, \mathbf{b}, \mathbf{c}) \)) is defined as \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \).

3. **Analyzing Each Case**:
- **Case 1**: Assume \( \alpha, \beta, \gamma \) are all non-zero. If \( \alpha, \beta, \gamma \neq 0 \), then the equation can be rearranged to show that the scalar triple product must equal zero for the equation to hold:
\[ \alpha (\mathbf{a} \times \mathbf{b}) + \beta (\mathbf{b} \times \mathbf{c}) + \gamma (\mathbf{c} \times \mathbf{a}) = 0 \]
implies that \( [\mathbf{a}, \mathbf{b}, \mathbf{c}] = 0 \), indicating that \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) are coplanar.

**Hint**: If the coefficients are all non-zero and the sum is zero, the vectors must be coplanar.

4. **Case 2**: If any one of \( \alpha, \beta, \gamma \) is non-zero, the same reasoning applies. For instance, if \( \alpha \neq 0 \), we can rearrange the equation to isolate the term involving \( \alpha \):
\[ \alpha (\mathbf{a} \times \mathbf{b}) = -(\beta (\mathbf{b} \times \mathbf{c}) + \gamma (\mathbf{c} \times \mathbf{a})) \]
This indicates that the left side must also yield a vector that is dependent on the right side, reinforcing that the vectors are coplanar.

**Hint**: If at least one coefficient is non-zero, the vectors remain dependent, and thus coplanar.

5. **Case 3**: If all coefficients are zero, the equation trivially holds, but this does not provide any information about the coplanarity of the vectors.

6. **Conclusion**:
- **Option A**: \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) are coplanar if all of \( \alpha, \beta, \gamma \) are non-zero. **Correct**.
- **Option B**: \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) are coplanar if any one of \( \alpha, \beta, \gamma \) is non-zero. **Correct**.
- **Option C**: \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) are non-coplanar for any \( \alpha, \beta, \gamma \) not equal to zero. **Incorrect**.
- **Option D**: None of these. **Incorrect**.

Thus, the correct options are A and B.

### Final Answer:
Options A and B are correct.