If `veca, vecb, vecc` are three non-coplanar vetors represented by non-current edges of a parallelopiped of volume 4 units, then the value of
`(veca+vecb).(vecbxxvecc)+(vecb+vecc).(veccxxveca)+(vecc+veca).(vecaxxvecb)`, is

Options

A. 12
B. 4
C. `+-12`
D. 0

Explanation

To solve the problem, we need to evaluate the expression:

\[ (\vec{a} + \vec{b}) \cdot (\vec{b} \times \vec{c}) + (\vec{b} + \vec{c}) \cdot (\vec{c} \times \vec{a}) + (\vec{c} + \vec{a}) \cdot (\vec{a} \times \vec{b}) \]

Given that \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) are non-coplanar vectors representing the edges of a parallelepiped with a volume of 4 units, we know that:

\[ |\vec{a} \cdot (\vec{b} \times \vec{c})| = 4 \]

### Step-by-Step Solution:

1. **Expand the first term**:
\[ (\vec{a} + \vec{b}) \cdot (\vec{b} \times \vec{c}) = \vec{a} \cdot (\vec{b} \times \vec{c}) + \vec{b} \cdot (\vec{b} \times \vec{c}) \]
Since \(\vec{b} \cdot (\vec{b} \times \vec{c}) = 0\) (the dot product of a vector with a vector perpendicular to it is zero), we have:
\[ (\vec{a} + \vec{b}) \cdot (\vec{b} \times \vec{c}) = \vec{a} \cdot (\vec{b} \times \vec{c}) \]

2. **Expand the second term**:
\[ (\vec{b} + \vec{c}) \cdot (\vec{c} \times \vec{a}) = \vec{b} \cdot (\vec{c} \times \vec{a}) + \vec{c} \cdot (\vec{c} \times \vec{a}) \]
Again, \(\vec{c} \cdot (\vec{c} \times \vec{a}) = 0\), so:
\[ (\vec{b} + \vec{c}) \cdot (\vec{c} \times \vec{a}) = \vec{b} \cdot (\vec{c} \times \vec{a}) \]

3. **Expand the third term**:
\[ (\vec{c} + \vec{a}) \cdot (\vec{a} \times \vec{b}) = \vec{c} \cdot (\vec{a} \times \vec{b}) + \vec{a} \cdot (\vec{a} \times \vec{b}) \]
Here, \(\vec{a} \cdot (\vec{a} \times \vec{b}) = 0\), thus:
\[ (\vec{c} + \vec{a}) \cdot (\vec{a} \times \vec{b}) = \vec{c} \cdot (\vec{a} \times \vec{b}) \]

4. **Combine all the terms**:
Now we can combine all the simplified terms:
\[ \vec{a} \cdot (\vec{b} \times \vec{c}) + \vec{b} \cdot (\vec{c} \times \vec{a}) + \vec{c} \cdot (\vec{a} \times \vec{b}) \]

5. **Use the scalar triple product**:
The expression \(\vec{a} \cdot (\vec{b} \times \vec{c})\) is the scalar triple product, which gives the volume of the parallelepiped formed by the vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\). Thus:
\[ \vec{a} \cdot (\vec{b} \times \vec{c}) = 4 \]
Similarly, we can show that:
\[ \vec{b} \cdot (\vec{c} \times \vec{a}) = 4 \quad \text{and} \quad \vec{c} \cdot (\vec{a} \times \vec{b}) = 4 \]

6. **Final calculation**:
Therefore, we have:
\[ 4 + 4 + 4 = 12 \]

### Final Answer:
The value of the expression is \(12\).