If `[(veca,vecb,vecc)]=3`, then the volume (in cubic units) of the parallelopiped with `2veca+vecb,2vecb+vecc` and `2vecc+veca` as coterminous edges is

Options

A. 15
B. 22
C. 25
D. 27

Explanation

To find the volume of the parallelepiped formed by the vectors \(2\vec{a} + \vec{b}\), \(2\vec{b} + \vec{c}\), and \(2\vec{c} + \vec{a}\), we can use the scalar triple product. The volume \(V\) of a parallelepiped defined by vectors \(\vec{u}\), \(\vec{v}\), and \(\vec{w}\) is given by:

\[ V = |\vec{u} \cdot (\vec{v} \times \vec{w})| \]

### Step-by-Step Solution:

1. **Define the vectors**:
Let:
\[ \vec{u} = 2\vec{a} + \vec{b} \]
\[ \vec{v} = 2\vec{b} + \vec{c} \]
\[ \vec{w} = 2\vec{c} + \vec{a} \]

2. **Compute the cross product**:
We need to find \(\vec{v} \times \vec{w}\):
\[ \vec{v} \times \vec{w} = (2\vec{b} + \vec{c}) \times (2\vec{c} + \vec{a}) \]

Using the distributive property of the cross product:
\[ \vec{v} \times \vec{w} = 2\vec{b} \times 2\vec{c} + 2\vec{b} \times \vec{a} + \vec{c} \times 2\vec{c} + \vec{c} \times \vec{a} \]
Since \(\vec{c} \times \vec{c} = \vec{0}\):
\[ \vec{v} \times \vec{w} = 4(\vec{b} \times \vec{c}) + 2(\vec{b} \times \vec{a}) + (\vec{c} \times \vec{a}) \]

3. **Compute the dot product**:
Now, we compute \(\vec{u} \cdot (\vec{v} \times \vec{w})\):
\[ \vec{u} \cdot (\vec{v} \times \vec{w}) = (2\vec{a} + \vec{b}) \cdot \left(4(\vec{b} \times \vec{c}) + 2(\vec{b} \times \vec{a}) + (\vec{c} \times \vec{a})\right) \]

Expanding this:
\[ = 2\vec{a} \cdot (4(\vec{b} \times \vec{c})) + 2\vec{a} \cdot (2(\vec{b} \times \vec{a})) + 2\vec{a} \cdot (\vec{c} \times \vec{a}) + \vec{b} \cdot (4(\vec{b} \times \vec{c})) + \vec{b} \cdot (2(\vec{b} \times \vec{a})) + \vec{b} \cdot (\vec{c} \times \vec{a}) \]

Noting that \(\vec{a} \cdot (\vec{b} \times \vec{a}) = 0\) and \(\vec{b} \cdot (\vec{b} \times \vec{c}) = 0\):
\[ = 8(\vec{a} \cdot (\vec{b} \times \vec{c})) + 0 + 0 + 0 + 0 + \vec{b} \cdot (\vec{c} \times \vec{a}) \]
\[ = 8(\vec{a} \cdot (\vec{b} \times \vec{c})) + \vec{b} \cdot (\vec{c} \times \vec{a}) \]

The term \(\vec{b} \cdot (\vec{c} \times \vec{a})\) is also equal to \(\vec{a} \cdot (\vec{b} \times \vec{c})\) due to the properties of the scalar triple product. Thus:
\[ \vec{u} \cdot (\vec{v} \times \vec{w}) = 8(\vec{a} \cdot (\vec{b} \times \vec{c})) + (\vec{a} \cdot (\vec{b} \times \vec{c})) = 9(\vec{a} \cdot (\vec{b} \times \vec{c})) \]

4. **Substitute the given value**:
We know that \([\vec{a}, \vec{b}, \vec{c}] = 3\), which is equal to \(\vec{a} \cdot (\vec{b} \times \vec{c})\).
\[ \vec{u} \cdot (\vec{v} \times \vec{w}) = 9 \times 3 = 27 \]

5. **Conclusion**:
The volume of the parallelepiped is:
\[ V = |\vec{u} \cdot (\vec{v} \times \vec{w})| = 27 \text{ cubic units} \]