If `[veca vecbvecc]=2` find the volume of the parallelepiped whose co-teminus edges are `2veca +vecb, 2 vecb + vecc, 2 vecc + veca.`

Answer

To find the volume of the parallelepiped whose co-terminus edges are given as \(2\vec{a} + \vec{b}\), \(2\vec{b} + \vec{c}\), and \(2\vec{c} + \vec{a}\), we will use the scalar triple product formula. The volume \(V\) of a parallelepiped formed by vectors \(\vec{u}\), \(\vec{v}\), and \(\vec{w}\) is given by:

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

### Step 1: Identify the vectors
The vectors corresponding to the edges of the parallelepiped are:
- \(\vec{u} = 2\vec{a} + \vec{b}\)
- \(\vec{v} = 2\vec{b} + \vec{c}\)
- \(\vec{w} = 2\vec{c} + \vec{a}\)

### Step 2: Compute the scalar triple product
We need to compute the scalar triple product \( \vec{u} \cdot (\vec{v} \times \vec{w}) \).

\[ \vec{v} \times \vec{w} = (2\vec{b} + \vec{c}) \times (2\vec{c} + \vec{a}) \]

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 \(\vec{c} \times \vec{c} = \vec{0}\), we can simplify:

\[ \vec{v} \times \vec{w} = 4(\vec{b} \times \vec{c}) + 2(\vec{b} \times \vec{a}) + \vec{c} \times \vec{a} \]

### Step 3: Substitute back into the scalar triple product
Now substitute \(\vec{v} \times \vec{w}\) into the scalar triple product:

\[ \vec{u} \cdot (\vec{v} \times \vec{w}) = (2\vec{a} + \vec{b}) \cdot \left(4(\vec{b} \times \vec{c}) + 2(\vec{b} \times \vec{a}) + \vec{c} \times \vec{a}\right) \]

### Step 4: Distribute the dot product
Now we distribute the dot product:

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

### Step 5: Simplify using properties of the dot and cross products
Using the properties of the dot product (where the dot product of a vector with a cross product involving itself is zero):

- \( \vec{a} \cdot (\vec{b} \times \vec{a}) = 0 \)
- \( \vec{b} \cdot (\vec{b} \times \vec{c}) = 0 \)

Thus, we can simplify:

\[ = 8(\vec{a} \cdot (\vec{b} \times \vec{c})) + \vec{b} \cdot (\vec{c} \times \vec{a}) \]

### Step 6: Substitute the known value
Given that \([\vec{a} \vec{b} \vec{c}] = 2\):

\[ = 8 \cdot 2 + 0 = 16 \]

### Step 7: Calculate the volume
The volume \(V\) is:

\[ V = |16| = 16 \]

### Final Answer
The volume of the parallelepiped is \(16\) cubic units.

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