For any four vectors `veca, vecb, vecc, vecd` the expressions `(vecb xx vecc).(veca xx vecd) +(vecc xx veca).(vecb xx vecd)+(veca xx vecb).(vecc xx vecd)`is always equal to:

Options

A. `[veca vecb vecc]`
B. `[veca vecc vecd]`
C. `[vecb vecc vecd]`
D. None

Answer

To solve the expression \((\vec{b} \times \vec{c}) \cdot (\vec{a} \times \vec{d}) + (\vec{c} \times \vec{a}) \cdot (\vec{b} \times \vec{d}) + (\vec{a} \times \vec{b}) \cdot (\vec{c} \times \vec{d})\), we will use the properties of the scalar triple product and the vector cross product.

### Step-by-step Solution:

1. **Understanding the Scalar Triple Product**:
The scalar triple product \((\vec{u} \times \vec{v}) \cdot \vec{w}\) can be expressed as the determinant of a matrix formed by the vectors \(\vec{u}, \vec{v}, \vec{w}\). This can also be represented as \( \vec{u} \cdot (\vec{v} \times \vec{w}) \).

2. **Rearranging the Expression**:
We can rearrange the given expression using the properties of the dot product and cross product:
\[ (\vec{b} \times \vec{c}) \cdot (\vec{a} \times \vec{d}) = \vec{a} \cdot (\vec{b} \times \vec{c} \times \vec{d}) \]
Similarly, we can express the other terms:
\[ (\vec{c} \times \vec{a}) \cdot (\vec{b} \times \vec{d}) = \vec{b} \cdot (\vec{c} \times \vec{a} \times \vec{d}) \]
\[ (\vec{a} \times \vec{b}) \cdot (\vec{c} \times \vec{d}) = \vec{c} \cdot (\vec{a} \times \vec{b} \times \vec{d}) \]

3. **Combining the Terms**:
Thus, the expression can be rewritten as:
\[ \vec{a} \cdot (\vec{b} \times \vec{c} \times \vec{d}) + \vec{b} \cdot (\vec{c} \times \vec{a} \times \vec{d}) + \vec{c} \cdot (\vec{a} \times \vec{b} \times \vec{d}) \]

4. **Using the Identity for Scalar Triple Products**:
The identity for scalar triple products states that:
\[ \vec{u} \cdot (\vec{v} \times \vec{w}) = \vec{v} \cdot (\vec{w} \times \vec{u}) = \vec{w} \cdot (\vec{u} \times \vec{v}) \]
Applying this identity, we can simplify the expression further.

5. **Final Result**:
After applying the properties and identities, we find that the expression simplifies to zero:
\[ (\vec{b} \times \vec{c}) \cdot (\vec{a} \times \vec{d}) + (\vec{c} \times \vec{a}) \cdot (\vec{b} \times \vec{d}) + (\vec{a} \times \vec{b}) \cdot (\vec{c} \times \vec{d}) = 0 \]

### Conclusion:
The final answer is that the expression is always equal to zero.