The value of `(vec(A)+vec(B))xx(vec(A)-vec(B))` is

Options

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

Explanation

To solve the problem of finding the value of \((\vec{A} + \vec{B}) \times (\vec{A} - \vec{B})\), we can follow these steps:

### Step 1: Expand the Expression
We start by using the distributive property of the cross product:
\[ (\vec{A} + \vec{B}) \times (\vec{A} - \vec{B}) = \vec{A} \times \vec{A} - \vec{A} \times \vec{B} + \vec{B} \times \vec{A} - \vec{B} \times \vec{B} \]

### Step 2: Simplify the Terms
Next, we simplify each term:
- The cross product of any vector with itself is zero:
\[ \vec{A} \times \vec{A} = 0 \quad \text{and} \quad \vec{B} \times \vec{B} = 0 \]
- Therefore, we can eliminate these terms:
\[ 0 - \vec{A} \times \vec{B} + \vec{B} \times \vec{A} - 0 = -\vec{A} \times \vec{B} + \vec{B} \times \vec{A} \]

### Step 3: Use the Property of Cross Products
We know that the cross product is anti-commutative, meaning:
\[ \vec{B} \times \vec{A} = -(\vec{A} \times \vec{B}) \]
Thus, we can substitute:
\[ -\vec{A} \times \vec{B} - \vec{A} \times \vec{B} = -2(\vec{A} \times \vec{B}) \]

### Step 4: Final Result
So, the final result is:
\[ (\vec{A} + \vec{B}) \times (\vec{A} - \vec{B}) = -2(\vec{A} \times \vec{B}) \]

### Summary
The value of \((\vec{A} + \vec{B}) \times (\vec{A} - \vec{B})\) is \(-2(\vec{A} \times \vec{B})\).

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