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. `axxb`

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 1: Expand the Expression
Using the distributive property of the cross product, we can expand the expression:
\[ (\mathbf{a} + \mathbf{b}) \times (\mathbf{a} - \mathbf{b}) = \mathbf{a} \times \mathbf{a} - \mathbf{a} \times \mathbf{b} + \mathbf{b} \times \mathbf{a} - \mathbf{b} \times \mathbf{b} \]

### Step 2: Simplify Each Term
Now, we simplify each term in the expanded expression:
- \(\mathbf{a} \times \mathbf{a} = \mathbf{0}\) (the cross product of any vector with itself is zero)
- \(\mathbf{b} \times \mathbf{b} = \mathbf{0}\) (similarly, the cross product of \(\mathbf{b}\) with itself is also zero)

Thus, the expression simplifies to:
\[ \mathbf{0} - \mathbf{a} \times \mathbf{b} + \mathbf{b} \times \mathbf{a} - \mathbf{0} \]

### Step 3: Use the Anti-Symmetric Property of Cross Product
The cross product is anti-symmetric, which means:
\[ \mathbf{b} \times \mathbf{a} = -(\mathbf{a} \times \mathbf{b}) \]

Substituting this into our expression gives:
\[ -\mathbf{a} \times \mathbf{b} - \mathbf{a} \times \mathbf{b} = -2(\mathbf{a} \times \mathbf{b}) \]

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