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 the volume of the parallelepiped formed by the vectors \(\vec{a}, \vec{b}, \vec{c}\) is 4 units, we can relate this to the scalar triple product:

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

### Step 1: Break down the expression

We can expand each term in the expression:

1. \((\vec{a} + \vec{b}) \cdot (\vec{b} \times \vec{c}) = \vec{a} \cdot (\vec{b} \times \vec{c}) + \vec{b} \cdot (\vec{b} \times \vec{c})\)
2. \((\vec{b} + \vec{c}) \cdot (\vec{c} \times \vec{a}) = \vec{b} \cdot (\vec{c} \times \vec{a}) + \vec{c} \cdot (\vec{c} \times \vec{a})\)
3. \((\vec{c} + \vec{a}) \cdot (\vec{a} \times \vec{b}) = \vec{c} \cdot (\vec{a} \times \vec{b}) + \vec{a} \cdot (\vec{a} \times \vec{b})\)

### Step 2: Identify zero terms

Using the property of the dot product, we know that:

- \(\vec{b} \cdot (\vec{b} \times \vec{c}) = 0\) (a vector dotted with a cross product involving itself is zero)
- \(\vec{c} \cdot (\vec{c} \times \vec{a}) = 0\)
- \(\vec{a} \cdot (\vec{a} \times \vec{b}) = 0\)

Thus, the expression simplifies to:

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

### Step 3: Use the scalar triple product

The remaining terms can be expressed using the scalar triple product:

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

Thus, we have:

\[ \vec{a} \cdot (\vec{b} \times \vec{c}) + \vec{b} \cdot (\vec{c} \times \vec{a}) + \vec{c} \cdot (\vec{a} \times \vec{b}) = V + V + V = 3V \]

### Step 4: Substitute the volume

Since the volume \(V\) is given as 4 units, we substitute:

\[ 3V = 3 \times 4 = 12 \]

### Conclusion

The value of the expression is:

\[ \boxed{12} \]