Find the volume of the parallelepiped with the given vertices. (0, 0, 0),  (3, 0, 0),  (0, 4, 1),  (2, 0, 4),  (3, 4, 1),  (5, 0, 4),  (2, 4, 5),  (5, 4, 5)

To find the volume of the parallelepiped formed by the given vertices, we can use the scalar triple product of three vectors originating from the same vertex, typically the origin (0, 0, 0).

### Step 1: Identify the three vectors
Choose three vectors originating from the vertex (0, 0, 0):
1. **Vector \( \mathbf{a} \)** from (0, 0, 0) to (3, 0, 0):
   \[
   \mathbf{a} = \langle 3, 0, 0 \rangle
   \]
2. **Vector \( \mathbf{b} \)** from (0, 0, 0) to (0, 4, 1):
   \[
   \mathbf{b} = \langle 0, 4, 1 \rangle
   \]
3. **Vector \( \mathbf{c} \)** from (0, 0, 0) to (2, 0, 4):
   \[
   \mathbf{c} = \langle 2, 0, 4 \rangle
   \]

### Step 2: Compute the cross product of \( \mathbf{b} \) and \( \mathbf{c} \)
The cross product \( \mathbf{b} \times \mathbf{c} \) is calculated as follows:
\[
\mathbf{b} \times \mathbf{c} = \begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
0 & 4 & 1 \\
2 & 0 & 4
\end{vmatrix}
= \mathbf{i}(4 \cdot 4 - 1 \cdot 0) - \mathbf{j}(0 \cdot 4 - 1 \cdot 2) + \mathbf{k}(0 \cdot 0 - 4 \cdot 2)
\]
\[
\mathbf{b} \times \mathbf{c} = \langle 16, -2, -8 \rangle
\]

### Step 3: Compute the scalar triple product \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \)
Now, compute the dot product of \( \mathbf{a} \) and \( \mathbf{b} \times \mathbf{c} \):
\[
\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = \langle 3, 0, 0 \rangle \cdot \langle 16, -2, -8 \rangle = 3 \cdot 16 + 0 \cdot (-2) + 0 \cdot (-8) = 48
\]

### Step 4: Find the volume
The volume of the parallelepiped is the absolute value of the scalar triple product:
\[
\text{Volume} = | \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) | = | 48 | = 48
\]

So, the volume of the parallelepiped is **48 cubic units**.

Would you like further details or have any questions?

Here are five related questions for further exploration:
1. How does the volume formula change for a parallelepiped if it's not aligned with the coordinate axes?
2. What is the geometric interpretation of the scalar triple product?
3. How can you compute the volume of a parallelepiped if given different sets of vectors?
4. How does the cross product of two vectors relate to the area of a parallelogram?
5. Can you determine the volume of a tetrahedron using similar principles?

**Tip:** Always check that the vectors you choose originate from the same point when calculating the scalar triple product to avoid errors.

Learn how to find the volume of a parallelepiped using vector geometry and the scalar triple product. This problem involves calculating the volume based on given vertices (0, 0, 0), (3, 0, 0), (0, 4, 1), (2, 0, 4), (3, 4, 1), (5, 0, 4), (2, 4, 5), (5, 4, 5). Suitable for advanced high school students and those interested in vector geometry.

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