If the vectors `veca, vecb, vecc, vecd` are any four vectors, then `(vecaxxvecb).(veccxxvecd)` is equal to

Options

A. `veca.{(bxx(vecc xx vecd)}`
B. `(veca.vecc)(vecb.vecd)-(veca.vecd)(vecb.vecc)`
C. `{(vecaxxvecb)xxvecc}.vecd`
D. `(vecdxxvecc).(vecbxxveca)`

Answer

To solve the problem, we need to find the value of the expression \((\vec{a} \times \vec{b}) \cdot (\vec{c} \times \vec{d})\).

### Step-by-Step Solution:

1. **Understanding the Expression**:
The expression \((\vec{a} \times \vec{b}) \cdot (\vec{c} \times \vec{d})\) represents the dot product of two cross products.

2. **Using the Scalar Triple Product Identity**:
We can use the identity that relates the dot product of two cross products to a determinant:
\[ \vec{u} \cdot (\vec{v} \times \vec{w}) = \text{det}(\vec{u}, \vec{v}, \vec{w}) \]
Therefore, we can rewrite our expression as:
\[ (\vec{a} \times \vec{b}) \cdot (\vec{c} \times \vec{d}) = \text{det}(\vec{a}, \vec{b}, \vec{c}, \vec{d}) \]

3. **Applying the Determinant Property**:
The determinant of four vectors can be expressed as:
\[ \text{det}(\vec{a}, \vec{b}, \vec{c}, \vec{d}) = \text{det}(\vec{a}, \vec{b}, \vec{c}) \cdot \vec{d} \]
However, for our specific case, we can directly state that:
\[ (\vec{a} \times \vec{b}) \cdot (\vec{c} \times \vec{d}) = \text{det}(\vec{a}, \vec{b}, \vec{c}, \vec{d}) \]

4. **Final Result**:
Hence, the value of \((\vec{a} \times \vec{b}) \cdot (\vec{c} \times \vec{d})\) is equal to the scalar triple product of the vectors \(\vec{a}, \vec{b}, \vec{c}, \vec{d}\).

### Conclusion:
The final answer is:
\[ (\vec{a} \times \vec{b}) \cdot (\vec{c} \times \vec{d}) = \text{det}(\vec{a}, \vec{b}, \vec{c}, \vec{d}) \]