If `axx(bxxc)` is perpendicular to `(axxb)xxc`, we may have

Options

A. `(veca.vecc)|vecb|^(2)=(veca.vecb)(vecb.vecc)`
B. `veca.vecb=0`
C. `veca.vecc=0`
D. `vecb.vecc=0`

Explanation

To solve the problem, we need to analyze the condition given: \( \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) \) is perpendicular to \( (\mathbf{a} \times \mathbf{b}) \times \mathbf{c} \). This implies that the dot product of these two vectors is zero.

### Step-by-Step Solution

1. **Understand the Cross Product**:
The cross product of two vectors \( \mathbf{u} \) and \( \mathbf{v} \) gives a vector that is perpendicular to both \( \mathbf{u} \) and \( \mathbf{v} \). We will use the vector triple product identity:
\[ \mathbf{u} \times (\mathbf{v} \times \mathbf{w}) = (\mathbf{u} \cdot \mathbf{w}) \mathbf{v} - (\mathbf{u} \cdot \mathbf{v}) \mathbf{w} \]

2. **Apply the Triple Product Identity**:
For \( \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) \):
\[ \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \cdot \mathbf{c}) \mathbf{b} - (\mathbf{a} \cdot \mathbf{b}) \mathbf{c} \]

For \( (\mathbf{a} \times \mathbf{b}) \times \mathbf{c} \):
\[ (\mathbf{a} \times \mathbf{b}) \times \mathbf{c} = (\mathbf{c} \cdot \mathbf{b}) \mathbf{a} - (\mathbf{c} \cdot \mathbf{a}) \mathbf{b} \]

3. **Set Up the Dot Product**:
Since the two vectors are perpendicular, we can set up the equation:
\[ \left[ (\mathbf{a} \cdot \mathbf{c}) \mathbf{b} - (\mathbf{a} \cdot \mathbf{b}) \mathbf{c} \right] \cdot \left[ (\mathbf{c} \cdot \mathbf{b}) \mathbf{a} - (\mathbf{c} \cdot \mathbf{a}) \mathbf{b} \right] = 0 \]

4. **Expand the Dot Product**:
Expanding the dot product gives:
\[ (\mathbf{a} \cdot \mathbf{c})(\mathbf{c} \cdot \mathbf{b}) (\mathbf{b} \cdot \mathbf{a}) - (\mathbf{a} \cdot \mathbf{c})(\mathbf{c} \cdot \mathbf{a}) (\mathbf{b} \cdot \mathbf{b}) - (\mathbf{a} \cdot \mathbf{b})(\mathbf{c} \cdot \mathbf{b}) (\mathbf{b} \cdot \mathbf{c}) + (\mathbf{a} \cdot \mathbf{b})(\mathbf{c} \cdot \mathbf{a}) (\mathbf{b} \cdot \mathbf{b}) = 0 \]

5. **Rearranging Terms**:
Rearranging and simplifying the equation will lead us to conditions on the vectors \( \mathbf{a}, \mathbf{b}, \) and \( \mathbf{c} \).

6. **Conclusion**:
After simplification, we can conclude that one of the conditions for the vectors to satisfy the perpendicularity condition is:
- \( \mathbf{a} \cdot \mathbf{c} = 0 \) or \( \mathbf{b} \cdot \mathbf{c} = 0 \)

### Final Result
Thus, the conditions that satisfy the given problem are:
- \( \mathbf{a} \cdot \mathbf{c} = 0 \) (indicating \( \mathbf{a} \) is perpendicular to \( \mathbf{c} \))
- \( \mathbf{b} \cdot \mathbf{c} = 0 \) (indicating \( \mathbf{b} \) is perpendicular to \( \mathbf{c} \))