https://file.aiquickdraw.com/mathbot/file/1723968759698gnfknog7.jpg

Let's solve this new problem using the same method as before.

### Problem 1: Volume of the Parallelepiped

The vectors given are:

\[
\mathbf{v_1} = \begin{bmatrix} -1 \\ 0 \\ 0 \end{bmatrix}, \quad \mathbf{v_2} = \begin{bmatrix} 0 \\ -2 \\ 3 \end{bmatrix}, \quad \mathbf{v_3} = \begin{bmatrix} -8 \\ -4 \\ 8 \end{bmatrix}
\]

The volume \( V \) of the parallelepiped is given by the determinant of the matrix formed by these vectors:

\[
V = \left| \text{det} \begin{bmatrix} -1 & 0 & -8 \\ 0 & -2 & -4 \\ 0 & 3 & 8 \end{bmatrix} \right|
\]

### Problem 2: Volume of the Tetrahedron

The tetrahedron's volume will be:

\[
V_{\text{tetrahedron}} = \frac{1}{6} \times V_{\text{parallelepiped}}
\]

I'll compute these values now.### Results

1. **Volume of the Parallelepiped:** The volume is approximately \( 4.0 \) cubic units.
2. **Volume of the Tetrahedron:** The volume is approximately \( 0.67 \) cubic units.

If you need further explanation or have any additional questions, feel free to ask!

Here are five related questions:

1. How does changing the order of vectors in the matrix affect the volume calculation?
2. What is the effect on the volume if one of the vectors is scaled by a factor of 2?
3. How can you interpret a zero determinant in the context of vector volumes?
4. What is the relationship between the base area and height in the context of parallelepiped volume?
5. How would the calculations differ if the vectors were in four-dimensional space?

**Tip:** When dealing with vectors that include zero components, ensure to carefully compute the determinant, as it might simplify the process or lead to special cases.

Learn how to calculate the volume of a parallelepiped and its relationship to the volume of a tetrahedron using vector operations and determinants. Includes step-by-step solutions and mathematical formulas. Suitable for advanced high school students and beyond.

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