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To find the volume of the parallelepiped defined by the vectors \(\mathbf{u} = \langle 2, -8, 2 \rangle\), \(\mathbf{v} = \langle 0, 4, -2 \rangle\), and \(\mathbf{w} = \langle 1, 2, -4 \rangle\), we need to compute the scalar triple product \(\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w})\).

### Step 1: Compute \(\mathbf{v} \times \mathbf{w}\) (cross product of \(\mathbf{v}\) and \(\mathbf{w}\)).

\[
\mathbf{v} \times \mathbf{w} =
\begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
0 & 4 & -2 \\
1 & 2 & -4
\end{vmatrix}
\]

This determinant expands as follows:

\[
\mathbf{v} \times \mathbf{w} = \mathbf{i} \left( 4(-4) - (-2)(2) \right) - \mathbf{j} \left( 0(-4) - (-2)(1) \right) + \mathbf{k} \left( 0(2) - 4(1) \right)
\]

Simplifying each term:

\[
\mathbf{i}( -16 + 4 ) - \mathbf{j}( 0 + 2 ) + \mathbf{k}( 0 - 4 )
\]

\[
= \mathbf{i}(-12) - \mathbf{j}(2) + \mathbf{k}(-4)
\]

Thus, \(\mathbf{v} \times \mathbf{w} = \langle -12, -2, -4 \rangle\).

### Step 2: Compute \(\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w})\) (dot product of \(\mathbf{u}\) and the result from Step 1).

\[
\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}) = \langle 2, -8, 2 \rangle \cdot \langle -12, -2, -4 \rangle
\]

The dot product is calculated as:

\[
2(-12) + (-8)(-2) + 2(-4) = -24 + 16 - 8 = -16
\]

### Step 3: Take the absolute value of the result.

The volume of the parallelepiped is the absolute value of the scalar triple product:

\[
\text{Volume} = | -16 | = 16
\]

Thus, the volume of the parallelepiped is **16 cubic units**.

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Do you have any questions or need further clarification? 

Here are some related questions for further exploration:
1. What is the geometric meaning of the cross product in the context of parallelepipeds?
2. How do you compute the cross product of two 3D vectors step-by-step?
3. What would happen to the volume if one of the vectors is a scalar multiple of another?
4. How can the scalar triple product help determine if three vectors are coplanar?
5. How would you find the area of the parallelogram formed by two vectors in 3D?

**Tip:** The scalar triple product's result being zero means the vectors are coplanar, implying no volume.

Find the volume of the parallelepiped that has u = ⟨2, -8, 2⟩, v = ⟨0, 4, -2⟩, and w = ⟨1, 2, -4⟩ as the adjacent sides.

Learn how to find the volume of a parallelepiped using the scalar triple product of three vectors: u = ⟨2, -8, 2⟩, v = ⟨0, 4, -2⟩, and w = ⟨1, 2, -4⟩. This problem involves the cross product and dot product calculations, suitable for advanced high school or college-level students.

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