Let `vec(a),vec(b)` and `vec(c )` be three coterminous edges of a tetrahedron with volume `(1)/(3)` then volume of a parallelopiped whose non-parallel sides are given by `vec(a)xxvec(b),vec(b)xxvec(c)"and" vec(c ) xx vec(a)` is

Options

A. `(1)/(2)`
B. 1
C. 2
D. 4

Explanation

To solve the problem, we need to find the volume of a parallelepiped whose non-parallel sides are given by the vectors \(\vec{a} \times \vec{b}\), \(\vec{b} \times \vec{c}\), and \(\vec{c} \times \vec{a}\), given that the volume of a tetrahedron formed by the vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) is \(\frac{1}{3}\).

### Step-by-Step Solution:

1. **Volume of Tetrahedron**:
The volume \(V\) of a tetrahedron formed by vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) is given by:
\[ V = \frac{1}{6} |\vec{a} \cdot (\vec{b} \times \vec{c})| \]
Given that the volume is \(\frac{1}{3}\), we can set up the equation:
\[ \frac{1}{6} |\vec{a} \cdot (\vec{b} \times \vec{c})| = \frac{1}{3} \]

2. **Solving for the Scalar Triple Product**:
To find \(|\vec{a} \cdot (\vec{b} \times \vec{c})|\), we multiply both sides of the equation by 6:
\[ |\vec{a} \cdot (\vec{b} \times \vec{c})| = 2 \]

3. **Volume of Parallelepiped**:
The volume \(V_p\) of a parallelepiped formed by vectors \(\vec{u}\), \(\vec{v}\), and \(\vec{w}\) is given by:
\[ V_p = |\vec{u} \cdot (\vec{v} \times \vec{w})| \]
In our case, we need to find the volume of the parallelepiped formed by the vectors \(\vec{a} \times \vec{b}\), \(\vec{b} \times \vec{c}\), and \(\vec{c} \times \vec{a}\).

4. **Using the Vector Triple Product**:
We can express the volume of the parallelepiped as:
\[ V_p = |(\vec{a} \times \vec{b}) \cdot ((\vec{b} \times \vec{c}) \times (\vec{c} \times \vec{a}))| \]

5. **Applying the Vector Triple Product Identity**:
Using the identity \(\vec{x} \times (\vec{y} \times \vec{z}) = (\vec{x} \cdot \vec{z})\vec{y} - (\vec{x} \cdot \vec{y})\vec{z}\), we can simplify:
\[ (\vec{b} \times \vec{c}) \times (\vec{c} \times \vec{a}) = (\vec{b} \cdot \vec{a})\vec{c} - (\vec{b} \cdot \vec{c})\vec{a} \]

6. **Substituting Back**:
Now substituting this back into our volume expression:
\[ V_p = |(\vec{a} \times \vec{b}) \cdot ((\vec{b} \cdot \vec{a})\vec{c} - (\vec{b} \cdot \vec{c})\vec{a})| \]

7. **Calculating the Dot Product**:
The dot product can be calculated as:
\[ V_p = |(\vec{a} \times \vec{b}) \cdot (\vec{b} \cdot \vec{a})\vec{c} - (\vec{b} \cdot \vec{c})(\vec{a} \times \vec{b}) \cdot \vec{a}| \]
Since \((\vec{a} \times \vec{b}) \cdot \vec{a} = 0\), the second term vanishes.

8. **Final Volume Calculation**:
Thus, we have:
\[ V_p = |(\vec{a} \cdot \vec{b})|\cdot |\vec{c}| \]
Since we know \(|\vec{a} \cdot (\vec{b} \times \vec{c})| = 2\), we can conclude that:
\[ V_p = 4 \]

### Conclusion:
The volume of the parallelepiped is \(4\).