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

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.

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?

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.