Statement I: If a is perpendicular to b and c, then `axx(bxxc)=0` Statement II: if a is perpendicular to b and c, then `bxxc=0`

Options

A. Both statement I and statement II are correct and statement II is the correct explanation of statement I
B. both statement I and statement II are correct but statement II is not the correct explanation of statement I
C. Statement I is correct but statement II is incorrect
D. Statement II is correct but statement I is incorrect

Answer

To solve the problem, we need to analyze both statements given in the question.

### Step-by-Step Solution:

**Statement I:** If \( \mathbf{a} \) is perpendicular to \( \mathbf{b} \) and \( \mathbf{c} \), then \( \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = 0 \).

1. **Understanding the Cross Product:**
The vector triple product identity states that:
\[ \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \cdot \mathbf{c}) \mathbf{b} - (\mathbf{a} \cdot \mathbf{b}) \mathbf{c} \]

2. **Given Conditions:**
Since \( \mathbf{a} \) is perpendicular to both \( \mathbf{b} \) and \( \mathbf{c} \), we have:
\[ \mathbf{a} \cdot \mathbf{b} = 0 \quad \text{and} \quad \mathbf{a} \cdot \mathbf{c} = 0 \]

3. **Substituting into the Identity:**
Now substituting these values into the vector triple product identity:
\[ \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (0) \mathbf{b} - (0) \mathbf{c} = 0 \]

4. **Conclusion for Statement I:**
Therefore, \( \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = 0 \) is true.

**Statement II:** If \( \mathbf{a} \) is perpendicular to \( \mathbf{b} \) and \( \mathbf{c} \), then \( \mathbf{b} \times \mathbf{c} = 0 \).

1. **Analyzing the Statement:**
The statement claims that if \( \mathbf{a} \) is perpendicular to both \( \mathbf{b} \) and \( \mathbf{c} \), then the cross product \( \mathbf{b} \times \mathbf{c} \) must be zero.

2. **Understanding the Cross Product:**
The cross product \( \mathbf{b} \times \mathbf{c} = 0 \) if and only if \( \mathbf{b} \) and \( \mathbf{c} \) are parallel or one of them is the zero vector.

3. **Given Conditions:**
The fact that \( \mathbf{a} \) is perpendicular to \( \mathbf{b} \) and \( \mathbf{c} \) does not imply any relationship between \( \mathbf{b} \) and \( \mathbf{c} \). They could be neither parallel nor perpendicular.

4. **Conclusion for Statement II:**
Therefore, \( \mathbf{b} \times \mathbf{c} = 0 \) is not necessarily true. This statement is false.

### Final Conclusion:
- **Statement I is true.**
- **Statement II is false.**