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

Options

A. 0
B. `vec(a) xx vec(b)`
C. `2(vec(a) xx vec(b))`
D. `|a|^(2) + |b|^(2)`

Answer

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 Expression
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:
1. \(\vec{a} \times \vec{a} = \vec{0}\) (the cross product of any vector with itself is zero).
2. \(\vec{b} \times \vec{b} = \vec{0}\) (similarly, the cross product of \(\vec{b}\) with itself is also zero).

So, we can rewrite the expression:
\[ = \vec{0} + \vec{a} \times \vec{b} - \vec{b} \times \vec{a} + \vec{0} \]
This simplifies to:
\[ = \vec{a} \times \vec{b} - \vec{b} \times \vec{a} \]

### Step 3: Apply the Anti-Symmetry Property
Using the property of the cross product, which states that \(\vec{m} \times \vec{n} = -(\vec{n} \times \vec{m})\), we can rewrite \(-\vec{b} \times \vec{a}\) as:
\[ = \vec{a} \times \vec{b} + \vec{a} \times \vec{b} \]
This gives us:
\[ = 2(\vec{a} \times \vec{b}) \]

### Final Result
Thus, the final result of the expression \((\vec{a} - \vec{b}) \times (\vec{a} + \vec{b})\) is:
\[ \boxed{2(\vec{a} \times \vec{b})} \]