If three concurrent edges of a parallelopiped of volume `V` represent vectors `veca,vecb,vecc` then the volume of the parallelopiped whose three concurrent edges are the three concurrent diagonals of the three faces of the given parallelopiped is

Options

A. `V`
B. `2V`
C. `3V`
D. none of these

Explanation

To find the volume of the parallelepiped whose edges are the diagonals of the faces of the original parallelepiped defined by vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\), we can follow these steps:

### Step 1: Understand the Volume of the Original Parallelepiped
The volume \(V\) of the original parallelepiped formed by the vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) is given by the scalar triple product:
\[ V = |\vec{a} \cdot (\vec{b} \times \vec{c})| \]

### Step 2: Determine the Diagonal Vectors
The diagonals of the faces of the parallelepiped can be represented as:
- Diagonal of the face formed by \(\vec{a}\) and \(\vec{b}\): \(\vec{d_1} = \vec{a} + \vec{b}\)
- Diagonal of the face formed by \(\vec{b}\) and \(\vec{c}\): \(\vec{d_2} = \vec{b} + \vec{c}\)
- Diagonal of the face formed by \(\vec{c}\) and \(\vec{a}\): \(\vec{d_3} = \vec{c} + \vec{a}\)

### Step 3: Calculate the Volume of the New Parallelepiped
The volume \(V'\) of the new parallelepiped formed by the diagonals \(\vec{d_1}\), \(\vec{d_2}\), and \(\vec{d_3}\) can be calculated using the scalar triple product:
\[ V' = |\vec{d_1} \cdot (\vec{d_2} \times \vec{d_3})| \]

### Step 4: Substitute the Diagonal Vectors
Substituting the diagonal vectors into the volume formula:
\[ V' = |(\vec{a} + \vec{b}) \cdot ((\vec{b} + \vec{c}) \times (\vec{c} + \vec{a}))| \]

### Step 5: Expand the Cross Product
To compute \((\vec{b} + \vec{c}) \times (\vec{c} + \vec{a})\), we can use the distributive property of the cross product:
\[ (\vec{b} + \vec{c}) \times (\vec{c} + \vec{a}) = \vec{b} \times \vec{c} + \vec{b} \times \vec{a} + \vec{c} \times \vec{c} + \vec{c} \times \vec{a} \]
Since \(\vec{c} \times \vec{c} = \vec{0}\), we simplify to:
\[ \vec{b} \times \vec{c} + \vec{b} \times \vec{a} + \vec{c} \times \vec{a} \]

### Step 6: Substitute Back into the Volume Formula
Now substitute this back into the volume formula:
\[ V' = |(\vec{a} + \vec{b}) \cdot (\vec{b} \times \vec{c} + \vec{b} \times \vec{a} + \vec{c} \times \vec{a})| \]

### Step 7: Calculate the Result
After performing the dot products and simplifying, we find that:
\[ V' = \frac{1}{2} |\vec{a} \cdot (\vec{b} \times \vec{c})| + \frac{1}{2} |\vec{b} \cdot (\vec{c} \times \vec{a})| + \frac{1}{2} |\vec{c} \cdot (\vec{a} \times \vec{b})| = \frac{3}{2} V \]

### Conclusion
Thus, the volume of the parallelepiped formed by the diagonals of the original parallelepiped is:
\[ V' = \frac{1}{2} V \]