`[(vec(a) xx vec(b)) xx (vec(b) xx vec(c)), (vec(b) xx vec(c)) xx (vec(c) xx vec(a)),(vec(c) xx vec(a)) xx (vec(a) xx vec(b))]` is equal to

Options

A. `[vec(a) vec(b) vec(c)]^(2)`
B. `[vec(a) vec(b)vec(c)]^(3)`
C. `[vec(a)vec(b)vec(c)]^(4)`
D. 0

Answer

To solve the problem, we need to evaluate the expression:

\[ [(\vec{a} \times \vec{b}) \times (\vec{b} \times \vec{c}), (\vec{b} \times \vec{c}) \times (\vec{c} \times \vec{a}), (\vec{c} \times \vec{a}) \times (\vec{a} \times \vec{b})] \]

This expression consists of three vector triple products. We will simplify each term using the vector triple product identity:

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

### Step 1: Simplifying the first term

The first term is \((\vec{a} \times \vec{b}) \times (\vec{b} \times \vec{c})\).

Using the vector triple product identity, we can write:

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

### Step 2: Simplifying the second term

The second term is \((\vec{b} \times \vec{c}) \times (\vec{c} \times \vec{a})\).

Applying the vector triple product identity again:

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

### Step 3: Simplifying the third term

The third term is \((\vec{c} \times \vec{a}) \times (\vec{a} \times \vec{b})\).

Using the vector triple product identity once more:

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

### Step 4: Combining the results

Now we have:

1. First term: \((\vec{a} \cdot \vec{c}) \vec{b} - (\vec{a} \cdot \vec{b}) \vec{c}\)
2. Second term: \((\vec{b} \cdot \vec{a}) \vec{c} - (\vec{b} \cdot \vec{c}) \vec{a}\)
3. Third term: \((\vec{c} \cdot \vec{b}) \vec{a} - (\vec{c} \cdot \vec{a}) \vec{b}\)

Now, we can combine these three results into a single expression:

\[ \begin{align*} & \left[(\vec{a} \cdot \vec{c}) \vec{b} - (\vec{a} \cdot \vec{b}) \vec{c}\right] + \left[(\vec{b} \cdot \vec{a}) \vec{c} - (\vec{b} \cdot \vec{c}) \vec{a}\right] + \left[(\vec{c} \cdot \vec{b}) \vec{a} - (\vec{c} \cdot \vec{a}) \vec{b}\right] \\ & = \left[(\vec{a} \cdot \vec{c}) \vec{b} + (\vec{b} \cdot \vec{a}) \vec{c} + (\vec{c} \cdot \vec{b}) \vec{a}\right] - \left[(\vec{a} \cdot \vec{b}) \vec{c} + (\vec{b} \cdot \vec{c}) \vec{a} + (\vec{c} \cdot \vec{a}) \vec{b}\right] \end{align*} \]

### Final Result

The final result can be expressed as:

\[ \text{The expression is equal to } 0 \]

This is because each term cancels out with its corresponding negative term.