If `veca , vecb and vecc` are non- coplanar vectors and `veca xx vecc` is perpendicular to `veca xx (vecb xx vecc)` , then the value of `[ veca xx ( vecb xx vecc)] xx vecc` is equal to

Options

A. ` [veca vecb vecc] vecc`
B. `[ veca vecb vecc] vecb`
C. ` vec0`
D. `[veca vecb vecc] veca`

Explanation

To solve the problem, we need to find the value of \([ \vec{a} \times ( \vec{b} \times \vec{c}) ] \times \vec{c}\) given that \(\vec{a} \times \vec{c}\) is perpendicular to \(\vec{a} \times (\vec{b} \times \vec{c})\).

### Step-by-Step Solution:

1. **Understanding the Given Condition**:
We know that \(\vec{a} \times \vec{c}\) is perpendicular to \(\vec{a} \times (\vec{b} \times \vec{c})\). This means:
\[ (\vec{a} \times \vec{c}) \cdot (\vec{a} \times (\vec{b} \times \vec{c})) = 0 \]

2. **Using the Vector Triple Product Identity**:
We can use the vector triple product identity:
\[ \vec{u} \times (\vec{v} \times \vec{w}) = (\vec{u} \cdot \vec{w}) \vec{v} - (\vec{u} \cdot \vec{v}) \vec{w} \]
Here, let \(\vec{u} = \vec{a}\), \(\vec{v} = \vec{b}\), and \(\vec{w} = \vec{c}\). Thus:
\[ \vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c}) \vec{b} - (\vec{a} \cdot \vec{b}) \vec{c} \]

3. **Substituting into the Dot Product**:
Now substituting this into our dot product condition:
\[ (\vec{a} \times \vec{c}) \cdot \left[ (\vec{a} \cdot \vec{c}) \vec{b} - (\vec{a} \cdot \vec{b}) \vec{c} \right] = 0 \]
This expands to:
\[ (\vec{a} \times \vec{c}) \cdot ((\vec{a} \cdot \vec{c}) \vec{b}) - (\vec{a} \times \vec{c}) \cdot ((\vec{a} \cdot \vec{b}) \vec{c}) = 0 \]

4. **Analyzing Each Term**:
The first term can be simplified:
- Since \(\vec{a} \times \vec{c}\) is perpendicular to \(\vec{a}\), the first term becomes zero:
\[ (\vec{a} \cdot \vec{c}) \cdot 0 = 0 \]
The second term becomes:
\[ -(\vec{a} \cdot \vec{b}) (\vec{a} \times \vec{c}) \cdot \vec{c} \]
Since \(\vec{a} \times \vec{c}\) is perpendicular to \(\vec{c}\), this term also becomes zero:
\[ -(\vec{a} \cdot \vec{b}) \cdot 0 = 0 \]

5. **Conclusion**:
Since both terms equal zero, we conclude that:
\[ (\vec{a} \cdot \vec{c}) = 0 \quad \text{and} \quad (\vec{a} \cdot \vec{b}) = 0 \]
Hence, we can conclude that:
\[ [\vec{a} \times (\vec{b} \times \vec{c})] \times \vec{c} = \vec{0} \]

### Final Answer:
\[ [\vec{a} \times (\vec{b} \times \vec{c})] \times \vec{c} = \vec{0} \]