If `veca, vecb, vecc` are any three non coplanar vectors, then
`(veca+vecb+vecc).(vecb+vecc)xx(vecc+veca)`

Options

A. `0`
B. `[(veca, vecb, vecc)]`
C. `2[(veca, vecb, vecc)]`
D. `3[(veca, vecb, vecc)]`

Explanation

To solve the problem, we need to evaluate the expression \((\vec{a} + \vec{b} + \vec{c}) \cdot ((\vec{b} + \vec{c}) \times (\vec{c} + \vec{a}))\).

### Step-by-step Solution:

1. **Start with the expression**:
\[ (\vec{a} + \vec{b} + \vec{c}) \cdot ((\vec{b} + \vec{c}) \times (\vec{c} + \vec{a})) \]

2. **Expand the cross product**:
We need to compute \((\vec{b} + \vec{c}) \times (\vec{c} + \vec{a})\). Using the distributive property of the cross product, we have:
\[ (\vec{b} + \vec{c}) \times (\vec{c} + \vec{a}) = \vec{b} \times \vec{c} + \vec{b} \times \vec{a} + \vec{c} \times \vec{c} + \vec{c} \times \vec{a} \]
Since \(\vec{c} \times \vec{c} = \vec{0}\) (the cross product of any vector with itself is zero), this simplifies to:
\[ \vec{b} \times \vec{c} + \vec{b} \times \vec{a} + \vec{c} \times \vec{a} \]

3. **Substitute back into the dot product**:
Now substitute this back into the original expression:
\[ (\vec{a} + \vec{b} + \vec{c}) \cdot (\vec{b} \times \vec{c} + \vec{b} \times \vec{a} + \vec{c} \times \vec{a}) \]

4. **Distribute the dot product**:
We can distribute the dot product:
\[ = (\vec{a} + \vec{b} + \vec{c}) \cdot (\vec{b} \times \vec{c}) + (\vec{a} + \vec{b} + \vec{c}) \cdot (\vec{b} \times \vec{a}) + (\vec{a} + \vec{b} + \vec{c}) \cdot (\vec{c} \times \vec{a}) \]

5. **Evaluate each term**:
- The first term: \((\vec{a} + \vec{b} + \vec{c}) \cdot (\vec{b} \times \vec{c})\) can be broken down:
- \(\vec{a} \cdot (\vec{b} \times \vec{c}) + \vec{b} \cdot (\vec{b} \times \vec{c}) + \vec{c} \cdot (\vec{b} \times \vec{c})\)
- The second and third terms are zero since the dot product of a vector with a vector perpendicular to it is zero. Thus, this term simplifies to:
\[ \vec{a} \cdot (\vec{b} \times \vec{c}) \]

- The second term: \((\vec{a} + \vec{b} + \vec{c}) \cdot (\vec{b} \times \vec{a})\) simplifies similarly:
- \(\vec{a} \cdot (\vec{b} \times \vec{a}) + \vec{b} \cdot (\vec{b} \times \vec{a}) + \vec{c} \cdot (\vec{b} \times \vec{a})\)
- The first and second terms are zero, so this simplifies to:
\[ \vec{c} \cdot (\vec{b} \times \vec{a}) \]

- The third term: \((\vec{a} + \vec{b} + \vec{c}) \cdot (\vec{c} \times \vec{a})\) simplifies to:
- \(\vec{a} \cdot (\vec{c} \times \vec{a}) + \vec{b} \cdot (\vec{c} \times \vec{a}) + \vec{c} \cdot (\vec{c} \times \vec{a})\)
- The first and third terms are zero, so this simplifies to:
\[ \vec{b} \cdot (\vec{c} \times \vec{a}) \]

6. **Combine the results**:
Now we combine the results from all three terms:
\[ \vec{a} \cdot (\vec{b} \times \vec{c}) + \vec{c} \cdot (\vec{b} \times \vec{a}) + \vec{b} \cdot (\vec{c} \times \vec{a}) \]

7. **Recognize the scalar triple product**:
The expression we have is the scalar triple product:
\[ \vec{a} \cdot (\vec{b} \times \vec{c}) = \vec{b} \cdot (\vec{c} \times \vec{a}) = \vec{c} \cdot (\vec{a} \times \vec{b}) \]
Hence, the final result is:
\[ \text{Result} = \vec{a} \cdot (\vec{b} \times \vec{c}) = \text{Volume of the parallelepiped formed by } \vec{a}, \vec{b}, \vec{c} \]

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