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

Options

A. `vec(0)`
B. `vec(a) xx vec(b)`
C. `2 (vec(a) xx vec(b))`
D. `|vec(a)|^(2)- |vec(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. Here are the steps:

### Step 1: Expand the expression using the distributive property of the cross product.
We can expand the expression as follows:
\[ (\vec{a} - \vec{b}) \times (\vec{a} + \vec{b}) = \vec{a} \times \vec{a} + \vec{a} \times \vec{b} - \vec{b} \times \vec{a} - \vec{b} \times \vec{b} \]

### Step 2: Simplify the terms.
1. The cross product of any vector with itself is zero:
\[ \vec{a} \times \vec{a} = \vec{0} \]
\[ \vec{b} \times \vec{b} = \vec{0} \]
Therefore, we can simplify the expression:
\[ \vec{0} + \vec{a} \times \vec{b} - \vec{b} \times \vec{a} + \vec{0} = \vec{a} \times \vec{b} - \vec{b} \times \vec{a} \]

### Step 3: Use the anti-commutative property of the cross product.
The cross product is anti-commutative, meaning:
\[ \vec{b} \times \vec{a} = -(\vec{a} \times \vec{b}) \]
Thus, we can rewrite the expression:
\[ \vec{a} \times \vec{b} - (-\vec{a} \times \vec{b}) = \vec{a} \times \vec{b} + \vec{a} \times \vec{b} = 2(\vec{a} \times \vec{b}) \]

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

### Conclusion:
The answer is \(2 \vec{a} \times \vec{b}\), which corresponds to option C.

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