If `veca, vecb and vecc` are three non-coplanar vectors, then `(veca + vecb + vecc). [(veca + vecb) xx (veca + vecc)]` equals

Options

A. 0
B. `[veca vecb vecc]`
C. `2[veca vecb vecc]`
D. `-[veca vecb vecc]`

Explanation

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

### Step-by-Step Solution:

1. **Expand the Cross Product**:
We first need to compute the cross product \((\vec{a} + \vec{b}) \times (\vec{a} + \vec{c})\).
\[ (\vec{a} + \vec{b}) \times (\vec{a} + \vec{c}) = \vec{a} \times \vec{a} + \vec{a} \times \vec{c} + \vec{b} \times \vec{a} + \vec{b} \times \vec{c} \]
Since \(\vec{a} \times \vec{a} = \vec{0}\), we can simplify this to:
\[ \vec{a} \times \vec{c} + \vec{b} \times \vec{a} + \vec{b} \times \vec{c} \]

2. **Substitute Back into the Dot Product**:
Now, substitute this result back into the original expression:
\[ (\vec{a} + \vec{b} + \vec{c}) \cdot (\vec{a} \times \vec{c} + \vec{b} \times \vec{a} + \vec{b} \times \vec{c}) \]

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

4. **Evaluate Each Term**:
- The first term:
\[ (\vec{a} + \vec{b} + \vec{c}) \cdot (\vec{a} \times \vec{c}) = \vec{a} \cdot (\vec{a} \times \vec{c}) + \vec{b} \cdot (\vec{a} \times \vec{c}) + \vec{c} \cdot (\vec{a} \times \vec{c}) = 0 + \vec{b} \cdot (\vec{a} \times \vec{c}) + 0 = \vec{b} \cdot (\vec{a} \times \vec{c}) \]
- The second term:
\[ (\vec{a} + \vec{b} + \vec{c}) \cdot (\vec{b} \times \vec{a}) = \vec{a} \cdot (\vec{b} \times \vec{a}) + \vec{b} \cdot (\vec{b} \times \vec{a}) + \vec{c} \cdot (\vec{b} \times \vec{a}) = 0 + 0 + \vec{c} \cdot (\vec{b} \times \vec{a}) = \vec{c} \cdot (\vec{b} \times \vec{a}) \]
- The third term:
\[ (\vec{a} + \vec{b} + \vec{c}) \cdot (\vec{b} \times \vec{c}) = \vec{a} \cdot (\vec{b} \times \vec{c}) + \vec{b} \cdot (\vec{b} \times \vec{c}) + \vec{c} \cdot (\vec{b} \times \vec{c}) = \vec{a} \cdot (\vec{b} \times \vec{c}) + 0 + 0 = \vec{a} \cdot (\vec{b} \times \vec{c}) \]

5. **Combine the Results**:
Now, combine all the results:
\[ \vec{b} \cdot (\vec{a} \times \vec{c}) + \vec{c} \cdot (\vec{b} \times \vec{a}) + \vec{a} \cdot (\vec{b} \times \vec{c}) \]
This expression is known as the scalar triple product, which can be represented as:
\[ \text{Scalar Triple Product} = \vec{a} \cdot (\vec{b} \times \vec{c}) \]

6. **Final Result**:
Therefore, the final result is:
\[ = \text{Scalar Triple Product} - \text{Scalar Triple Product} = -2 \cdot \text{Scalar Triple Product} \]

### Conclusion:
The final answer is:
\[ -2 \cdot (\vec{a} \cdot (\vec{b} \times \vec{c})) \]