`( axxb) xx (a xxc ).d` equals

Options

A. `[abc][b.d]`
B. `[abc](a.d)`
C. `[abc](c.d)`
D. none of these

Answer

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

### Step-by-Step Solution:

1. **Identify the Expression**:
We start with the expression \((\mathbf{a} \times \mathbf{b}) \times (\mathbf{a} \times \mathbf{c}) \cdot \mathbf{d}\).

2. **Use the Vector Triple Product Identity**:
The vector triple product identity states that:
\[ \mathbf{x} \times (\mathbf{y} \times \mathbf{z}) = (\mathbf{x} \cdot \mathbf{z}) \mathbf{y} - (\mathbf{x} \cdot \mathbf{y}) \mathbf{z} \]
Here, we can let \(\mathbf{x} = \mathbf{a} \times \mathbf{b}\), \(\mathbf{y} = \mathbf{a}\), and \(\mathbf{z} = \mathbf{c}\).

3. **Apply the Identity**:
We apply the identity:
\[ (\mathbf{a} \times \mathbf{b}) \times (\mathbf{a} \times \mathbf{c}) = ((\mathbf{a} \times \mathbf{b}) \cdot \mathbf{c}) \mathbf{a} - ((\mathbf{a} \times \mathbf{b}) \cdot \mathbf{a}) \mathbf{c} \]

4. **Evaluate the Dot Products**:
- The term \((\mathbf{a} \times \mathbf{b}) \cdot \mathbf{a}\) is zero because the cross product \(\mathbf{a} \times \mathbf{b}\) is perpendicular to \(\mathbf{a}\).
- Therefore, we have:
\[ (\mathbf{a} \times \mathbf{b}) \times (\mathbf{a} \times \mathbf{c}) = ((\mathbf{a} \times \mathbf{b}) \cdot \mathbf{c}) \mathbf{a} \]

5. **Dot Product with \(\mathbf{d}\)**:
Now, we take the dot product of the result with \(\mathbf{d}\):
\[ ((\mathbf{a} \times \mathbf{b}) \cdot \mathbf{c}) \mathbf{a} \cdot \mathbf{d} \]

6. **Final Expression**:
The final expression simplifies to:
\[ (\mathbf{a} \times \mathbf{b}) \cdot \mathbf{c} \cdot (\mathbf{a} \cdot \mathbf{d}) \]

7. **Conclusion**:
The result is a scalar value given by:
\[ (\mathbf{a} \times \mathbf{b}) \cdot \mathbf{c} \cdot (\mathbf{a} \cdot \mathbf{d}) \]

### Final Answer:
The expression \((\mathbf{a} \times \mathbf{b}) \times (\mathbf{a} \times \mathbf{c}) \cdot \mathbf{d}\) equals \((\mathbf{a} \times \mathbf{b}) \cdot \mathbf{c} \cdot (\mathbf{a} \cdot \mathbf{d})\).