Find the value of `(vec(a) + vec(b)) xx (vec(a) - vec(b))`=

Options

A. `(vec(a) xx vec(b))`
B. `2(vec(a) xx vec(b))`
C. `-2(vec(a).vec(b))`
D. `-2 (vec(a) xx vec(b))`

Answer

To solve the expression \((\vec{a} + \vec{b}) \times (\vec{a} - \vec{b})\), we can expand it using the distributive property of the cross product.

### Step-by-Step Solution:

1. **Expand the Expression**:
\[ (\vec{a} + \vec{b}) \times (\vec{a} - \vec{b}) = \vec{a} \times \vec{a} - \vec{a} \times \vec{b} + \vec{b} \times \vec{a} - \vec{b} \times \vec{b} \]

2. **Evaluate the Cross Products**:
- The cross product of any vector with itself is zero:
\[ \vec{a} \times \vec{a} = \vec{0} \]
\[ \vec{b} \times \vec{b} = \vec{0} \]
- Therefore, we can simplify the expression:
\[ \vec{0} - \vec{a} \times \vec{b} + \vec{b} \times \vec{a} - \vec{0} = -\vec{a} \times \vec{b} + \vec{b} \times \vec{a} \]

3. **Use the Property of Cross Products**:
- We know that \(\vec{b} \times \vec{a} = -(\vec{a} \times \vec{b})\). Thus, we can substitute:
\[ -\vec{a} \times \vec{b} - \vec{a} \times \vec{b} = -2(\vec{a} \times \vec{b}) \]

4. **Final Result**:
\[ (\vec{a} + \vec{b}) \times (\vec{a} - \vec{b}) = -2(\vec{a} \times \vec{b}) \]

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

---