`(a + b) xx (a-b)` is equal to

Options

A. `a^(2) - b^(2)`
B. `2(a xx b)`
C. `2(b xxa)`
D. none of these

Answer

To solve the expression \((\mathbf{a} + \mathbf{b}) \times (\mathbf{a} - \mathbf{b})\), we will use the properties of the cross product and distribute the terms. Here’s the step-by-step solution:

### Step 1: Distribute 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 Cross Products
Next, we know that the cross product of any vector with itself is the zero vector:
\[ \mathbf{a} \times \mathbf{a} = \mathbf{0} \quad \text{and} \quad \mathbf{b} \times \mathbf{b} = \mathbf{0} \]
Thus, we can simplify the expression:
\[ \mathbf{0} - \mathbf{a} \times \mathbf{b} + \mathbf{b} \times \mathbf{a} - \mathbf{0} = -\mathbf{a} \times \mathbf{b} + \mathbf{b} \times \mathbf{a} \]

### Step 3: Use the Anti-Symmetry Property
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 result of the expression \((\mathbf{a} + \mathbf{b}) \times (\mathbf{a} - \mathbf{b})\) is:
\[ \boxed{-2\mathbf{a} \times \mathbf{b}} \]

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