The volume of a parallelepiped determined by three vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ is given by the absolute value of the scalar triple product:

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

Step 1: Compute the cross product of $\mathbf{b} \times \mathbf{c}$

Let: $\mathbf{b} = \begin{pmatrix} -1 \\ 1 \\ 4 \end{pmatrix}, \quad \mathbf{c} = \begin{pmatrix} 5 \\ 1 \\ 4 \end{pmatrix}$

The cross product $\mathbf{b} \times \mathbf{c}$ is calculated as follows:

$\mathbf{b} \times \mathbf{c} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ -1 & 1 & 4 \\ 5 & 1 & 4 \end{vmatrix}$

This determinant expands to:

$\mathbf{b} \times \mathbf{c} = \mathbf{i} \begin{vmatrix} 1 & 4 \\ 1 & 4 \end{vmatrix} - \mathbf{j} \begin{vmatrix} -1 & 4 \\ 5 & 4 \end{vmatrix} + \mathbf{k} \begin{vmatrix} -1 & 1 \\ 5 & 1 \end{vmatrix}$

Calculating each 2x2 determinant:

• $\mathbf{i}$-component: $1 \times 4 - 1 \times 4 = 0$
• $\mathbf{j}$-component: $-1 \times 4 - 5 \times 4 = -4 - 20 = -24$
• $\mathbf{k}$-component: $-1 \times 1 - 5 \times 1 = -1 - 5 = -6$

Thus, the cross product is:

$\mathbf{b} \times \mathbf{c} = \begin{pmatrix} 0 \\ -24 \\ -6 \end{pmatrix}$

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

Now, let: $\mathbf{a} = \begin{pmatrix} 1 \\ 4 \\ 2 \end{pmatrix}$

The dot product is:

$\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = 1 \times 0 + 4 \times (-24) + 2 \times (-6)$

$\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = 0 - 96 - 12 = -108$

Step 3: Take the absolute value

$V = | -108 | = 108 \text{ cubic units}$

Thus, the volume of the parallelepiped is $\boxed{108}$ cubic units.

Would you like more details or have any questions?

Here are some related problems:

1. Find the volume of a parallelepiped with different vectors.
2. Calculate the scalar triple product for another set of vectors.
3. Compute the cross product of two given vectors.
4. Solve for the determinant of a 3x3 matrix.
5. Find the area of a parallelogram formed by two vectors.

Tip: The scalar triple product $\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})$ is a useful tool to determine whether vectors are coplanar — if the result is zero, they are coplanar.