If a,b and c are three non-coplanar vectors, then the scalar product of vectors `a xx b + b xx c + c xx a` and `a + b + c ` is

Options

A. `[abc]`
B. `2[abc]`
C. `3[abc]`
D. none

Answer

To solve the problem, we need to find the scalar product of the vector \( \mathbf{a} \times \mathbf{b} + \mathbf{b} \times \mathbf{c} + \mathbf{c} \times \mathbf{a} \) with the vector \( \mathbf{a} + \mathbf{b} + \mathbf{c} \).

### Step-by-Step Solution:

1. **Identify the Vectors**:
Let \( \mathbf{A} = \mathbf{a} + \mathbf{b} + \mathbf{c} \) and \( \mathbf{B} = \mathbf{a} \times \mathbf{b} + \mathbf{b} \times \mathbf{c} + \mathbf{c} \times \mathbf{a} \).

2. **Calculate the Scalar Product**:
We need to compute \( \mathbf{A} \cdot \mathbf{B} \):
\[ \mathbf{A} \cdot \mathbf{B} = (\mathbf{a} + \mathbf{b} + \mathbf{c}) \cdot (\mathbf{a} \times \mathbf{b} + \mathbf{b} \times \mathbf{c} + \mathbf{c} \times \mathbf{a}) \]

3. **Distribute the Dot Product**:
Using the distributive property of the dot product:
\[ \mathbf{A} \cdot \mathbf{B} = \mathbf{a} \cdot (\mathbf{a} \times \mathbf{b}) + \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) + \mathbf{a} \cdot (\mathbf{c} \times \mathbf{a}) + \mathbf{b} \cdot (\mathbf{a} \times \mathbf{b}) + \mathbf{b} \cdot (\mathbf{b} \times \mathbf{c}) + \mathbf{b} \cdot (\mathbf{c} \times \mathbf{a}) + \mathbf{c} \cdot (\mathbf{a} \times \mathbf{b}) + \mathbf{c} \cdot (\mathbf{b} \times \mathbf{c}) + \mathbf{c} \cdot (\mathbf{c} \times \mathbf{a}) \]

4. **Evaluate Each Term**:
- The scalar product \( \mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}) \) is zero if \( \mathbf{u} \) is one of the vectors \( \mathbf{v} \) or \( \mathbf{w} \).
- Therefore:
- \( \mathbf{a} \cdot (\mathbf{a} \times \mathbf{b}) = 0 \)
- \( \mathbf{a} \cdot (\mathbf{c} \times \mathbf{a}) = 0 \)
- \( \mathbf{b} \cdot (\mathbf{b} \times \mathbf{c}) = 0 \)
- \( \mathbf{c} \cdot (\mathbf{c} \times \mathbf{a}) = 0 \)

5. **Remaining Terms**:
The remaining terms are:
- \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \)
- \( \mathbf{b} \cdot (\mathbf{c} \times \mathbf{a}) \)
- \( \mathbf{c} \cdot (\mathbf{a} \times \mathbf{b}) \)

By the properties of scalar triple products:
\[ \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = \mathbf{b} \cdot (\mathbf{c} \times \mathbf{a}) = \mathbf{c} \cdot (\mathbf{a} \times \mathbf{b}) = V \]
where \( V \) is the volume of the parallelepiped formed by \( \mathbf{a}, \mathbf{b}, \mathbf{c} \).

6. **Combine the Results**:
Therefore, the total scalar product becomes:
\[ \mathbf{A} \cdot \mathbf{B} = V + V + V = 3V \]

### Final Answer:
The scalar product of the vectors \( \mathbf{a} \times \mathbf{b} + \mathbf{b} \times \mathbf{c} + \mathbf{c} \times \mathbf{a} \) and \( \mathbf{a} + \mathbf{b} + \mathbf{c} \) is \( 3(\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})) \) or \( 3V \).