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

Options

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

Answer

To solve the problem, we need to find the value of the expression \((\vec{a} + \vec{b}) \times (\vec{a} - \vec{b})\).

### Step-by-Step Solution:

1. **Apply the Cross Product Distributive Property**:
We can expand the expression using the distributive property of the cross product:
\[ (\vec{a} + \vec{b}) \times (\vec{a} - \vec{b}) = \vec{a} \times \vec{a} - \vec{a} \times \vec{b} + \vec{b} \times \vec{a} - \vec{b} \times \vec{b} \]

2. **Evaluate the Cross Products**:
- The cross product of any vector with itself is zero:
\[ \vec{a} \times \vec{a} = \vec{0} \]
\[ \vec{b} \times \vec{b} = \vec{0} \]
- Thus, 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**:
The cross product is anti-commutative, meaning:
\[ \vec{b} \times \vec{a} = -(\vec{a} \times \vec{b}) \]
Therefore, we can substitute:
\[ -\vec{a} \times \vec{b} - \vec{a} \times \vec{b} = -2\vec{a} \times \vec{b} \]

4. **Final Result**:
Thus, the value of the expression \((\vec{a} + \vec{b}) \times (\vec{a} - \vec{b})\) is:
\[ -2\vec{a} \times \vec{b} \]

### Summary:
The final answer is:
\[ (\vec{a} + \vec{b}) \times (\vec{a} - \vec{b}) = -2\vec{a} \times \vec{b} \]