The question asks to calculate the scalar triple product of three vectors $\mathbf{a}, \mathbf{b}, \mathbf{c}$, where:

$\mathbf{a} = \{4, -4, 1\}, \quad \mathbf{b} = \{-1, 1, 3\}, \quad \mathbf{c} = \{2, -3, 2\}$

The scalar triple product is given by:

$\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})$

Step 1: Calculate $\mathbf{b} \times \mathbf{c}$ (cross product)

The cross product $\mathbf{b} \times \mathbf{c}$ can be calculated using the determinant formula:

$\mathbf{b} \times \mathbf{c} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ -1 & 1 & 3 \\ 2 & -3 & 2 \end{vmatrix}$

Expanding the determinant:

$\mathbf{b} \times \mathbf{c} = \mathbf{i} \left( \begin{vmatrix} 1 & 3 \\ -3 & 2 \end{vmatrix} \right) - \mathbf{j} \left( \begin{vmatrix} -1 & 3 \\ 2 & 2 \end{vmatrix} \right) + \mathbf{k} \left( \begin{vmatrix} -1 & 1 \\ 2 & -3 \end{vmatrix} \right)$

This simplifies to:

$\mathbf{b} \times \mathbf{c} = \mathbf{i} \left( 1(2) - 3(-3) \right) - \mathbf{j} \left( (-1)(2) - 3(2) \right) + \mathbf{k} \left( (-1)(-3) - 1(2) \right)$ $= \mathbf{i}(2 + 9) - \mathbf{j}(-2 - 6) + \mathbf{k}(3 - 2)$ $= 11\mathbf{i} + 8\mathbf{j} + 1\mathbf{k}$ Thus, $\mathbf{b} \times \mathbf{c} = \{11, 8, 1\}$.

Step 2: Calculate $\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})$ (dot product)

Now, calculate the dot product:

$\mathbf{a} \cdot \{11, 8, 1\} = 4(11) + (-4)(8) + 1(1)$ $= 44 - 32 + 1 = 13$

The scalar triple product is $13$.

Final Answer:

The correct answer is option 1: $13$.

Would you like more details on this calculation or have any further questions?

Here are five questions related to the scalar triple product:

1. What is the geometric interpretation of the scalar triple product?
2. How can you determine if three vectors are coplanar using the scalar triple product?
3. How does the cross product differ from the dot product in terms of vector operations?
4. What is the significance of a positive versus a negative result in the scalar triple product?
5. Can the scalar triple product result in zero? Under what conditions?

Tip: The scalar triple product represents the volume of the parallelepiped formed by three vectors.