https://file.aiquickdraw.com/mathbot/file/17303120018139mh6f2lc.jpg

Let's analyze and solve each question in the image.

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### Question 3: Volume of the Pyramid
The vertices of the triangular pyramid are given as:
- \( A(0, -3, 1) \)
- \( B(-4, 1, 2) \)
- \( C(2, -1, 5) \)
- \( D(3, 1, -4) \)

To find the volume \( V \) of a tetrahedron with vertices \( A \), \( B \), \( C \), and \( D \), we can use the formula:

\[
V = \frac{1}{6} \left| \text{det} \begin{pmatrix} x_B - x_A & y_B - y_A & z_B - z_A \\ x_C - x_A & y_C - y_A & z_C - z_A \\ x_D - x_A & y_D - y_A & z_D - z_A \end{pmatrix} \right|
\]

Substitute the coordinates for each point into the determinant and evaluate to find the volume.

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### Question 4: Area of the Triangle
The vertices of the triangle are given as:
- \( A(4, 1, -2) \)
- \( B(6, 3, 7) \)
- \( C(2, 3, 1) \)

To find the area \( A \) of the triangle with vertices \( A \), \( B \), and \( C \) in 3D space, we can use the formula:

\[
A = \frac{1}{2} \| \overrightarrow{AB} \times \overrightarrow{AC} \|
\]

1. Find vectors \( \overrightarrow{AB} = B - A \) and \( \overrightarrow{AC} = C - A \).
2. Compute the cross product \( \overrightarrow{AB} \times \overrightarrow{AC} \).
3. Find the magnitude of the cross product and divide by 2 to get the area.

Would you like me to compute these steps in detail for each question?

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### Relative Questions

1. How is the cross product useful in finding areas in 3D geometry?
2. What is the geometric meaning of a determinant in 3D space?
3. How does the formula for a tetrahedron's volume change in 2D?
4. What other methods exist for calculating volumes in 3D space?
5. Why does the volume formula include a \( \frac{1}{6} \) factor?

### Tip

When working with 3D coordinates, always double-check vector directions to ensure cross products and determinants are set up correctly.

Let A, B, C, D be the vertices of the triangular pyramid. Find the volume of the pyramid. Points: A(0, -3, 1), B(-4, 1, 2), C(2, -1, 5), D(3, 1, -4).

Let A, B, C be the vertices of the triangle. Find the area of the triangle. Points: A(4, 1, -2), B(6, 3, 7), C(2, 3, 1).

Learn how to calculate the volume of a tetrahedron and the area of a triangle in 3D space using determinants and cross products. Detailed step-by-step solutions for the points A(0, -3, 1), B(-4, 1, 2), C(2, -1, 5), D(3, 1, -4) for the pyramid, and A(4, 1, -2), B(6, 3, 7), C(2, 3, 1) for the triangle.

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