`vecaxx(vecaxx(vecaxxvecb))` equals

Options

A. `(veca.vecb)(vecaxxvecb)`
B. `(veca.veca)(vecbxxveca)`
C. `(vecb.vecb)(vecaxxvecb)`
D. `(vecb.vecb)(vecbxxveca)`

Explanation

To solve the expression \( \vec{a} \times (\vec{a} \times (\vec{a} \times \vec{b})) \), we will use the vector triple product identity. The identity states that:

\[ \vec{x} \times (\vec{y} \times \vec{z}) = (\vec{x} \cdot \vec{z}) \vec{y} - (\vec{x} \cdot \vec{y}) \vec{z} \]

### Step 1: Apply the triple product identity

Let \( \vec{y} = \vec{a} \) and \( \vec{z} = \vec{b} \). We first need to evaluate \( \vec{a} \times (\vec{a} \times \vec{b}) \).

Using the identity:

\[ \vec{a} \times (\vec{a} \times \vec{b}) = (\vec{a} \cdot \vec{b}) \vec{a} - (\vec{a} \cdot \vec{a}) \vec{b} \]

### Step 2: Substitute the result back into the original expression

Now we substitute this result back into the original expression:

\[ \vec{a} \times (\vec{a} \times (\vec{a} \times \vec{b})) = \vec{a} \times \left( (\vec{a} \cdot \vec{b}) \vec{a} - (\vec{a} \cdot \vec{a}) \vec{b} \right) \]

### Step 3: Distribute the cross product

Using the distributive property of the cross product, we have:

\[ \vec{a} \times \left( (\vec{a} \cdot \vec{b}) \vec{a} \right) - \vec{a} \times \left( (\vec{a} \cdot \vec{a}) \vec{b} \right) \]

### Step 4: Evaluate each term

1. The first term \( \vec{a} \times \left( (\vec{a} \cdot \vec{b}) \vec{a} \right) \) is zero because the cross product of any vector with itself is zero:

\[ \vec{a} \times \vec{a} = 0 \]

2. The second term can be simplified as follows:

\[ \vec{a} \times \left( (\vec{a} \cdot \vec{a}) \vec{b} \right) = (\vec{a} \cdot \vec{a}) (\vec{a} \times \vec{b}) \]

### Step 5: Combine the results

Putting it all together, we find:

\[ \vec{a} \times (\vec{a} \times (\vec{a} \times \vec{b})) = 0 - (\vec{a} \cdot \vec{a}) (\vec{a} \times \vec{b}) = - (\vec{a} \cdot \vec{a}) (\vec{a} \times \vec{b}) \]

### Final Result

Thus, the final result is:

\[ \vec{a} \times (\vec{a} \times (\vec{a} \times \vec{b})) = - (\vec{a} \cdot \vec{a}) (\vec{a} \times \vec{b}) \]