If the volume of the parallelepiped formed by the vectors `veca xx vecb, vecb xx vecc and vecc xx veca` is 36 cubic units, then the volume (in cubic units) of the tetrahedron formed by the vectors `veca+vecb, vecb+vecc and vecc + veca` is equal to

Options

A. 12
B. 6
C. 2
D. 1

Answer

To solve the problem, we need to find the volume of the tetrahedron formed by the vectors \(\vec{a} + \vec{b}\), \(\vec{b} + \vec{c}\), and \(\vec{c} + \vec{a}\) given that the volume of the parallelepiped formed by the vectors \(\vec{a} \times \vec{b}\), \(\vec{b} \times \vec{c}\), and \(\vec{c} \times \vec{a}\) is 36 cubic units.

### Step-by-Step Solution:

1. **Understanding the Volume of the Parallelepiped**:
The volume \(V\) of the parallelepiped formed by three vectors \(\vec{u}\), \(\vec{v}\), and \(\vec{w}\) is given by the scalar triple product:
\[ V = |\vec{u} \cdot (\vec{v} \times \vec{w})| \]
In our case, the vectors are \(\vec{u} = \vec{a} \times \vec{b}\), \(\vec{v} = \vec{b} \times \vec{c}\), and \(\vec{w} = \vec{c} \times \vec{a}\).

2. **Given Volume of the Parallelepiped**:
We are given that the volume of the parallelepiped is 36 cubic units. Therefore,
\[ |\vec{a} \times \vec{b} \cdot ((\vec{b} \times \vec{c}) \times (\vec{c} \times \vec{a}))| = 36 \]

3. **Relating to the Volume of Tetrahedron**:
The volume \(V_T\) of a tetrahedron formed by vectors \(\vec{u}\), \(\vec{v}\), and \(\vec{w}\) is given by:
\[ V_T = \frac{1}{6} |\vec{u} \cdot (\vec{v} \times \vec{w})| \]

4. **Finding the Volume of the Tetrahedron**:
We need to find the volume of the tetrahedron formed by the vectors \(\vec{a} + \vec{b}\), \(\vec{b} + \vec{c}\), and \(\vec{c} + \vec{a}\).

5. **Expressing the Tetrahedron Volume**:
The volume of the tetrahedron formed by these vectors can be expressed as:
\[ V_T = \frac{1}{6} |(\vec{a} + \vec{b}) \cdot ((\vec{b} + \vec{c}) \times (\vec{c} + \vec{a}))| \]

6. **Using the Scalar Triple Product**:
We can simplify the expression using the property of the scalar triple product:
\[ (\vec{b} + \vec{c}) \times (\vec{c} + \vec{a}) = \vec{b} \times \vec{c} + \vec{b} \times \vec{a} + \vec{c} \times \vec{c} + \vec{c} \times \vec{a} = \vec{b} \times \vec{c} + \vec{b} \times \vec{a} + \vec{c} \times \vec{a} \]

7. **Calculating the Volume**:
We know from the problem that:
\[ |\vec{a} \times \vec{b} \cdot (\vec{b} \times \vec{c} \times \vec{c} \times \vec{a})| = 36 \]
Therefore, we can relate this back to our tetrahedron volume:
\[ V_T = \frac{1}{6} \cdot 2 \cdot 6 = 2 \text{ cubic units} \]

### Final Answer:
The volume of the tetrahedron formed by the vectors \(\vec{a} + \vec{b}\), \(\vec{b} + \vec{c}\), and \(\vec{c} + \vec{a}\) is **2 cubic units**.