If a and b are two vectors.then the value of `(a+b)xx(a-b)` is

Options

A. `2(bxxa)`
B. `-2(bxxa)`
C. `bxxa`
D. `bxxa`

Explanation

To solve the problem of finding the value of \((\mathbf{a} + \mathbf{b}) \times (\mathbf{a} - \mathbf{b})\), we can use the properties of the cross product. Let's break it down step by step.

### Step-by-Step Solution:

1. **Write the Expression**:
We start with the expression:
\[ (\mathbf{a} + \mathbf{b}) \times (\mathbf{a} - \mathbf{b}) \]

2. **Apply the Distributive Property**:
Using the distributive property of the cross product, we can expand the expression:
\[ = \mathbf{a} \times \mathbf{a} - \mathbf{a} \times \mathbf{b} + \mathbf{b} \times \mathbf{a} - \mathbf{b} \times \mathbf{b} \]

3. **Evaluate Each Term**:
- The cross product of any vector with itself is zero:
\[ \mathbf{a} \times \mathbf{a} = \mathbf{0} \]
\[ \mathbf{b} \times \mathbf{b} = \mathbf{0} \]
- Thus, we can simplify:
\[ = \mathbf{0} - \mathbf{a} \times \mathbf{b} + \mathbf{b} \times \mathbf{a} - \mathbf{0} \]

4. **Use the Anti-symmetry Property**:
The cross product is anti-symmetric, meaning:
\[ \mathbf{b} \times \mathbf{a} = -(\mathbf{a} \times \mathbf{b}) \]
Therefore, we can substitute:
\[ = -\mathbf{a} \times \mathbf{b} - \mathbf{a} \times \mathbf{b} \]
\[ = -2(\mathbf{a} \times \mathbf{b}) \]

5. **Final Result**:
Thus, the final value of \((\mathbf{a} + \mathbf{b}) \times (\mathbf{a} - \mathbf{b})\) is:
\[ = -2(\mathbf{a} \times \mathbf{b}) \]

### Conclusion:
The value of \((\mathbf{a} + \mathbf{b}) \times (\mathbf{a} - \mathbf{b})\) is \(-2(\mathbf{a} \times \mathbf{b})\).

---