If `axxb=bxxc ne0`, then the correct statement is

Options

A. `b||c`
B. `a||b`
C. `(a+c)||b`
D. none of these

Answer

To solve the problem, we start with the given condition:

If \( \mathbf{a} \times \mathbf{b} = \mathbf{b} \times \mathbf{c} \neq \mathbf{0} \), we need to analyze the implications of this equation.

### Step-by-Step Solution:

1. **Understanding Cross Products**:
We know that the cross product of two vectors is a vector that is perpendicular to both of the original vectors. If \( \mathbf{a} \times \mathbf{b} = \mathbf{b} \times \mathbf{c} \), it implies that both cross products yield the same vector.

2. **Using Properties of Cross Products**:
The property of the cross product states that \( \mathbf{u} \times \mathbf{v} = -(\mathbf{v} \times \mathbf{u}) \). Therefore, we can rewrite:
\[ \mathbf{a} \times \mathbf{b} = -(\mathbf{c} \times \mathbf{b}) \]
This means:
\[ \mathbf{a} \times \mathbf{b} + \mathbf{c} \times \mathbf{b} = \mathbf{0} \]

3. **Factoring Out the Common Vector**:
We can factor out \( \mathbf{b} \) from the equation:
\[ \mathbf{b} \times (\mathbf{a} + \mathbf{c}) = \mathbf{0} \]

4. **Interpreting the Result**:
The equation \( \mathbf{b} \times (\mathbf{a} + \mathbf{c}) = \mathbf{0} \) implies that \( \mathbf{b} \) is parallel to \( \mathbf{a} + \mathbf{c} \). This is because the cross product of two vectors is zero if and only if the vectors are parallel.

5. **Conclusion**:
Therefore, we conclude that:
\[ \mathbf{a} + \mathbf{c} \text{ is parallel to } \mathbf{b} \]
This corresponds to option C in the question.

### Final Answer:
The correct statement is that \( \mathbf{a} + \mathbf{c} \) is parallel to \( \mathbf{b} \).

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