What is `(vec(a)-vec(b))xx(vec(a)+vec(b))` equal to?

Options

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

Explanation

To solve the expression \((\vec{a} - \vec{b}) \times (\vec{a} + \vec{b})\), we will use the properties of the cross product. Let's break it down step by step.

### Step 1: Expand the Cross Product
We start with the expression:
\[ (\vec{a} - \vec{b}) \times (\vec{a} + \vec{b}) \]

Using the distributive property of the cross product, we can expand this as follows:
\[ \vec{a} \times \vec{a} + \vec{a} \times \vec{b} - \vec{b} \times \vec{a} - \vec{b} \times \vec{b} \]

### Step 2: Simplify the Terms
Now, we simplify each term:
- The cross product of any vector with itself is zero:
\[ \vec{a} \times \vec{a} = 0 \quad \text{and} \quad \vec{b} \times \vec{b} = 0 \]
Thus, the expression simplifies to:
\[ 0 + \vec{a} \times \vec{b} - \vec{b} \times \vec{a} + 0 \]
This reduces to:
\[ \vec{a} \times \vec{b} - \vec{b} \times \vec{a} \]

### Step 3: Use the Antisymmetry Property
We know that the cross product is antisymmetric, which means:
\[ \vec{b} \times \vec{a} = -(\vec{a} \times \vec{b}) \]
Substituting this into our expression gives:
\[ \vec{a} \times \vec{b} - (-\vec{a} \times \vec{b}) = \vec{a} \times \vec{b} + \vec{a} \times \vec{b} \]
This simplifies to:
\[ 2(\vec{a} \times \vec{b}) \]

### Final Result
Therefore, the final result is:
\[ (\vec{a} - \vec{b}) \times (\vec{a} + \vec{b}) = 2(\vec{a} \times \vec{b}) \]