The value of `veca.(vecb+vecc)xx(veca+vecb+vecc)`, is

Options

A. `2[veca vecb vecc]`
B. `[veca vecb vecc]`
C. 0
D. None of these

Explanation

To solve the problem \( \vec{a} \cdot (\vec{b} + \vec{c}) \times (\vec{a} + \vec{b} + \vec{c}) \), we will follow these steps:

### Step 1: Expand the expression
We start by expanding the expression using the distributive property of the dot product and the cross product.

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

### Step 2: Calculate the cross products
Now, we will calculate the cross products separately.

1. **Calculate \( \vec{b} \times (\vec{a} + \vec{b} + \vec{c}) \)**:
\[ \vec{b} \times (\vec{a} + \vec{b} + \vec{c}) = \vec{b} \times \vec{a} + \vec{b} \times \vec{b} + \vec{b} \times \vec{c} \]
Since \( \vec{b} \times \vec{b} = \vec{0} \), we have:
\[ \vec{b} \times (\vec{a} + \vec{b} + \vec{c}) = \vec{b} \times \vec{a} + \vec{b} \times \vec{c} \]

2. **Calculate \( \vec{c} \times (\vec{a} + \vec{b} + \vec{c}) \)**:
\[ \vec{c} \times (\vec{a} + \vec{b} + \vec{c}) = \vec{c} \times \vec{a} + \vec{c} \times \vec{b} + \vec{c} \times \vec{c} \]
Since \( \vec{c} \times \vec{c} = \vec{0} \), we have:
\[ \vec{c} \times (\vec{a} + \vec{b} + \vec{c}) = \vec{c} \times \vec{a} + \vec{c} \times \vec{b} \]

### Step 3: Combine the results
Now we can combine the results from the two cross products:
\[ \vec{a} \cdot \left( \vec{b} \times \vec{a} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a} + \vec{c} \times \vec{b} \right) \]

### Step 4: Evaluate the dot products
Now we evaluate the dot products:
1. \( \vec{a} \cdot (\vec{b} \times \vec{a}) = 0 \) (since the dot product of a vector with itself is zero)
2. \( \vec{a} \cdot (\vec{c} \times \vec{c}) = 0 \) (same reason)
3. \( \vec{a} \cdot (\vec{b} \times \vec{c}) \) remains as it is.
4. \( \vec{a} \cdot (\vec{c} \times \vec{b}) = -\vec{a} \cdot (\vec{b} \times \vec{c}) \) (using the property of the scalar triple product)

### Step 5: Combine the results
Thus, we have:
\[ 0 + \vec{a} \cdot (\vec{b} \times \vec{c}) - \vec{a} \cdot (\vec{b} \times \vec{c}) = 0 \]

### Final Result
The value of \( \vec{a} \cdot (\vec{b} + \vec{c}) \times (\vec{a} + \vec{b} + \vec{c}) \) is:
\[ \boxed{0} \]