If `a xx b = c xx b ne 0`, then

Options

A. `a = lambdab`
B. `a-b=lambdac`
C. `(a-c) = lambdab`
D. none

Answer

To solve the problem, we need to analyze the equation given: \( A \times B = C \times B \neq 0 \).

### Step-by-Step Solution:

1. **Start with the given equation**:
\[ A \times B = C \times B \]

2. **Rearranging the equation**:
We can rearrange this equation to bring all terms to one side:
\[ A \times B - C \times B = 0 \]

3. **Factoring out \( B \)**:
From the left-hand side, we can factor out \( B \):
\[ (A - C) \times B = 0 \]

4. **Understanding the cross product**:
The equation \( (A - C) \times B = 0 \) implies that either:
- \( A - C = 0 \) (which means \( A = C \)), or
- \( B = 0 \), or
- \( A - C \) is parallel to \( B \).

5. **Analyzing the conditions**:
- If \( A - C = 0 \), then \( A = C \).
- If \( B = 0 \), it contradicts the given condition that \( C \times B \neq 0 \).
- If \( A - C \) is parallel to \( B \), it means there exists a scalar \( \lambda \) such that:
\[ A - C = \lambda B \]

6. **Conclusion**:
Therefore, the valid conditions derived from the original equation are:
- \( A = C \)
- \( A - C \) is parallel to \( B \)

### Final Answer:
The correct options based on the analysis are:
1. \( A = C \)
2. \( A - C \) is parallel to \( B \)