the value of `(vecA+vecB)xx(vecA-vecB)` is

Options

A. 0
B. `A^2-B^2`
C. `vecBxxvecA`
D. `2(vecBxxvecA)`

Explanation

To solve the problem of finding the value of \((\vec{A} + \vec{B}) \times (\vec{A} - \vec{B})\), we can use the distributive property of the cross product. Here’s a step-by-step breakdown of the solution:

### Step 1: Write the expression
We start with the expression we need to evaluate:
\[ (\vec{A} + \vec{B}) \times (\vec{A} - \vec{B}) \]

### Step 2: Apply the distributive property
Using the distributive property of the cross product, we can expand the expression:
\[ \vec{A} \times \vec{A} + \vec{A} \times (-\vec{B}) + \vec{B} \times \vec{A} + \vec{B} \times (-\vec{B}) \]
This simplifies to:
\[ \vec{A} \times \vec{A} - \vec{A} \times \vec{B} + \vec{B} \times \vec{A} - \vec{B} \times \vec{B} \]

### Step 3: Simplify the terms
Now, we know that the cross product of any vector with itself is zero:
\[ \vec{A} \times \vec{A} = \vec{0} \quad \text{and} \quad \vec{B} \times \vec{B} = \vec{0} \]
Thus, the expression reduces to:
\[ \vec{0} - \vec{A} \times \vec{B} + \vec{B} \times \vec{A} + \vec{0} \]
This simplifies further to:
\[ -\vec{A} \times \vec{B} + \vec{B} \times \vec{A} \]

### Step 4: Use the anti-commutative property
The cross product is anti-commutative, meaning:
\[ \vec{B} \times \vec{A} = -(\vec{A} \times \vec{B}) \]
Substituting this into our expression gives:
\[ -\vec{A} \times \vec{B} - \vec{A} \times \vec{B} = -2(\vec{A} \times \vec{B}) \]

### Final Result
Thus, the final value of \((\vec{A} + \vec{B}) \times (\vec{A} - \vec{B})\) is:
\[ -2(\vec{A} \times \vec{B}) \]