Find the volume of the parallelepiped determined by the vectors a, b, and c.
a = 

1, 4, 2

,
  b = 

−1, 1, 4

,
  c = 

5, 1, 4

 cubic units

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.

Find the volume of the parallelepiped determined by the vectors a, b, and c. a = (1, 4, 2), b = (−1, 1, 4), c = (5, 1, 4) cubic units

Learn how to compute the volume of a parallelepiped determined by three vectors: a = (1, 4, 2), b = (−1, 1, 4), and c = (5, 1, 4). This solution covers key vector algebra concepts, including the cross product, dot product, and scalar triple product. Suitable for advanced high school or undergraduate students.

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