If a, b and c three non-coplanar vectors, then `(a + b +c). [a+b) xx (a +c)` equals

Options

A. 0
B. `[abc]`
C. `2[abc]`
D. `-[abc]`

Answer

To solve the problem, we need to evaluate the expression \((\mathbf{a} + \mathbf{b} + \mathbf{c}) \cdot (\mathbf{a} + \mathbf{b}) \times (\mathbf{a} + \mathbf{c})\).

### Step-by-Step Solution:

1. **Understand the Expression**: We need to compute the dot product of the vector \((\mathbf{a} + \mathbf{b} + \mathbf{c})\) with the cross product \((\mathbf{a} + \mathbf{b}) \times (\mathbf{a} + \mathbf{c})\).

2. **Apply the Distributive Property**:
\[ (\mathbf{a} + \mathbf{b}) \times (\mathbf{a} + \mathbf{c}) = \mathbf{a} \times \mathbf{a} + \mathbf{a} \times \mathbf{c} + \mathbf{b} \times \mathbf{a} + \mathbf{b} \times \mathbf{c} \]
Since \(\mathbf{a} \times \mathbf{a} = \mathbf{0}\), we can simplify:
\[ (\mathbf{a} + \mathbf{b}) \times (\mathbf{a} + \mathbf{c}) = \mathbf{a} \times \mathbf{c} + \mathbf{b} \times \mathbf{a} + \mathbf{b} \times \mathbf{c} \]

3. **Substitute Back into the Dot Product**:
\[ (\mathbf{a} + \mathbf{b} + \mathbf{c}) \cdot (\mathbf{a} \times \mathbf{c} + \mathbf{b} \times \mathbf{a} + \mathbf{b} \times \mathbf{c}) \]

4. **Distribute the Dot Product**:
\[ = (\mathbf{a} + \mathbf{b} + \mathbf{c}) \cdot (\mathbf{a} \times \mathbf{c}) + (\mathbf{a} + \mathbf{b} + \mathbf{c}) \cdot (\mathbf{b} \times \mathbf{a}) + (\mathbf{a} + \mathbf{b} + \mathbf{c}) \cdot (\mathbf{b} \times \mathbf{c}) \]

5. **Evaluate Each Dot Product**:
- The first term: \((\mathbf{a} + \mathbf{b} + \mathbf{c}) \cdot (\mathbf{a} \times \mathbf{c})\) is zero because \(\mathbf{a} \cdot (\mathbf{a} \times \mathbf{c}) = 0\) and \(\mathbf{b} \cdot (\mathbf{a} \times \mathbf{c})\) is also zero.
- The second term: \((\mathbf{a} + \mathbf{b} + \mathbf{c}) \cdot (\mathbf{b} \times \mathbf{a})\) is zero for similar reasons.
- The third term: \((\mathbf{a} + \mathbf{b} + \mathbf{c}) \cdot (\mathbf{b} \times \mathbf{c})\) is also zero.

6. **Final Result**:
Since all terms evaluated to zero, we conclude:
\[ (\mathbf{a} + \mathbf{b} + \mathbf{c}) \cdot ((\mathbf{a} + \mathbf{b}) \times (\mathbf{a} + \mathbf{c})) = 0 \]

### Final Answer:
The value of \((\mathbf{a} + \mathbf{b} + \mathbf{c}) \cdot ((\mathbf{a} + \mathbf{b}) \times (\mathbf{a} + \mathbf{c}))\) is **0**.