If a is perpendicular to b and c, then

Options

A. `a xx (b xx c) =1`
B. `veca xx( vecbxx vecc) = vec0`
C. `a xx (b xx c) =-1`
D. none of these

Answer

To solve the problem, we need to analyze the relationship between the vectors \( \mathbf{a} \), \( \mathbf{b} \), and \( \mathbf{c} \) given that \( \mathbf{a} \) is perpendicular to both \( \mathbf{b} \) and \( \mathbf{c} \).

### Step-by-Step Solution:

1. **Understand the given information**:
We know that \( \mathbf{a} \perp \mathbf{b} \) and \( \mathbf{a} \perp \mathbf{c} \). This means that the dot product of \( \mathbf{a} \) with both \( \mathbf{b} \) and \( \mathbf{c} \) is zero:
\[ \mathbf{a} \cdot \mathbf{b} = 0 \quad \text{and} \quad \mathbf{a} \cdot \mathbf{c} = 0 \]

2. **Use the cross product**:
Since \( \mathbf{b} \) and \( \mathbf{c} \) are vectors, we can find the cross product \( \mathbf{b} \times \mathbf{c} \). The result of this cross product will be a vector that is perpendicular to both \( \mathbf{b} \) and \( \mathbf{c} \).

3. **Determine the relationship**:
Since \( \mathbf{a} \) is perpendicular to both \( \mathbf{b} \) and \( \mathbf{c} \), it must also be parallel to the vector resulting from \( \mathbf{b} \times \mathbf{c} \). This means:
\[ \mathbf{a} \parallel (\mathbf{b} \times \mathbf{c}) \]

4. **Formulate the cross product**:
If \( \mathbf{a} \) is parallel to \( \mathbf{b} \times \mathbf{c} \), then we can express this relationship using the cross product:
\[ \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = \mathbf{0} \]
This is because the cross product of two parallel vectors is the zero vector.

5. **Conclusion**:
Therefore, the final result is:
\[ \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = \mathbf{0} \]