If `veca, vecb, vecc` are three vectors, then
`[(vecaxxvecb, vecbxxvecc, veccxxveca)]=`

Options

A. `[(veca, vecb, vecc)]`
B. `2[(veca, vecb, vecc)]`
C. `3[(veca, vecb, vecc)]`
D. `[(veca, vecb, vecc)]^(2)`

Explanation

To solve the problem involving the scalar triple product of three vectors \( \vec{a}, \vec{b}, \vec{c} \), we need to evaluate the expression \( [(\vec{a} \times \vec{b}), (\vec{b} \times \vec{c}), (\vec{c} \times \vec{a})] \).

### Step-by-Step Solution:

1. **Understanding the Scalar Triple Product**:
The scalar triple product of three vectors \( \vec{a}, \vec{b}, \vec{c} \) can be expressed as:
\[ [\vec{a}, \vec{b}, \vec{c}] = \vec{a} \cdot (\vec{b} \times \vec{c}) \]
This represents the volume of the parallelepiped formed by the three vectors.

2. **Applying the Vector Triple Product Identity**:
We need to find the scalar triple product of the cross products:
\[ [(\vec{a} \times \vec{b}), (\vec{b} \times \vec{c}), (\vec{c} \times \vec{a})] \]
We can use the property of the scalar triple product, which states that the scalar triple product is invariant under cyclic permutations of the vectors.

3. **Using the Vector Triple Product Formula**:
We can express the scalar triple product using the vector triple product identity:
\[ \vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c}) \vec{b} - (\vec{a} \cdot \vec{b}) \vec{c} \]
We will apply this identity to our expression.

4. **Calculating Each Cross Product**:
- First, calculate \( \vec{a} \times \vec{b} \).
- Next, calculate \( \vec{b} \times \vec{c} \).
- Finally, calculate \( \vec{c} \times \vec{a} \).

5. **Combining the Results**:
Substitute the results of the cross products into the scalar triple product:
\[ [(\vec{a} \times \vec{b}), (\vec{b} \times \vec{c}), (\vec{c} \times \vec{a})] = \vec{a} \cdot ((\vec{b} \times \vec{c}) \times (\vec{c} \times \vec{a})) \]
Use the vector triple product identity to simplify this expression.

6. **Final Result**:
After applying the necessary simplifications and using the properties of the scalar triple product, we find that:
\[ [(\vec{a} \times \vec{b}), (\vec{b} \times \vec{c}), (\vec{c} \times \vec{a})] = (\vec{a} \cdot \vec{b} \times \vec{c})^2 \]
Therefore, the final answer is:
\[ [(\vec{a} \times \vec{b}), (\vec{b} \times \vec{c}), (\vec{c} \times \vec{a})] = (\vec{a} \cdot \vec{b} \times \vec{c})^2 \]