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, we need to find the value of \((\mathbf{a} + \mathbf{b}) \times (\mathbf{a} - \mathbf{b})\). Let's break this down step by step.

### Step 1: Apply the distributive property of the cross product
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 the terms
Now, let's simplify each term:

1. \(\mathbf{a} \times \mathbf{a} = \mathbf{0}\) (the cross product of any vector with itself is zero)
2. \(\mathbf{a} \times (-\mathbf{b}) = -(\mathbf{a} \times \mathbf{b})\)
3. \(\mathbf{b} \times \mathbf{a} = -(\mathbf{a} \times \mathbf{b})\) (the cross product is anti-commutative)
4. \(\mathbf{b} \times (-\mathbf{b}) = \mathbf{0}\)

Putting these together, we have:

\[ (\mathbf{a} + \mathbf{b}) \times (\mathbf{a} - \mathbf{b}) = \mathbf{0} - (\mathbf{a} \times \mathbf{b}) - (\mathbf{a} \times \mathbf{b}) + \mathbf{0} \]

### Step 3: Combine the results
Now, combine the results:

\[ = -\mathbf{a} \times \mathbf{b} - \mathbf{a} \times \mathbf{b} = -2(\mathbf{a} \times \mathbf{b}) \]

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

\[ \boxed{-2(\mathbf{a} \times \mathbf{b})} \]

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