If `veca xx (vec b xx vecc)` is perpendicular to `(veca xx vecb ) xx vecc`, 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 given condition that \(\vec{a} \times (\vec{b} \times \vec{c})\) is perpendicular to \((\vec{a} \times \vec{b}) \times \vec{c}\).

### Step-by-Step Solution:

1. **Understanding the Cross Product**:
The vector \(\vec{a} \times (\vec{b} \times \vec{c})\) can be rewritten using the vector triple product identity:
\[ \vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c}) \vec{b} - (\vec{a} \cdot \vec{b}) \vec{c} \]

2. **Rewriting the Second Vector**:
Similarly, we can rewrite \((\vec{a} \times \vec{b}) \times \vec{c}\) using the same identity:
\[ (\vec{a} \times \vec{b}) \times \vec{c} = (\vec{c} \cdot \vec{a}) \vec{b} - (\vec{c} \cdot \vec{b}) \vec{a} \]

3. **Setting Up the Perpendicular Condition**:
Since \(\vec{a} \times (\vec{b} \times \vec{c})\) is perpendicular to \((\vec{a} \times \vec{b}) \times \vec{c}\), their dot product must be zero:
\[ [(\vec{a} \cdot \vec{c}) \vec{b} - (\vec{a} \cdot \vec{b}) \vec{c}] \cdot [(\vec{c} \cdot \vec{a}) \vec{b} - (\vec{c} \cdot \vec{b}) \vec{a}] = 0 \]

4. **Expanding the Dot Product**:
Expanding this dot product yields:
\[ (\vec{a} \cdot \vec{c})(\vec{c} \cdot \vec{a}) \|\vec{b}\|^2 - (\vec{a} \cdot \vec{b})(\vec{c} \cdot \vec{b})(\vec{a} \cdot \vec{c}) - (\vec{a} \cdot \vec{c})(\vec{c} \cdot \vec{b})(\vec{a} \cdot \vec{b}) + (\vec{a} \cdot \vec{b})^2 (\vec{c} \cdot \vec{b}) = 0 \]

5. **Factoring the Expression**:
We can factor out \((\vec{a} \cdot \vec{c})\):
\[ (\vec{a} \cdot \vec{c}) \left[ \|\vec{b}\|^2 - (\vec{a} \cdot \vec{b})(\vec{c} \cdot \vec{b}) \right] = 0 \]

6. **Finding the Conditions**:
From the factored expression, we have two cases:
- Case 1: \(\vec{a} \cdot \vec{c} = 0\) (which means \(\vec{a}\) is perpendicular to \(\vec{c}\))
- Case 2: \(\|\vec{b}\|^2 = \vec{a} \cdot \vec{b} \cdot \vec{c} \cdot \vec{b}\)

7. **Conclusion**:
Therefore, the conditions we derived indicate that either \(\vec{a} \cdot \vec{c} = 0\) or the second condition holds. We can check which options match these conditions.