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To calculate the volume of a tetrahedron with vertices \( A(2, -3, 5) \), \( B(0, 2, 1) \), \( C(-2, -2, 3) \), and \( D(3, 2, 4) \), we use the determinant formula:

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

### Step 1: Define the vectors
- \( \vec{AB} = B - A = (0 - 2, 2 - (-3), 1 - 5) = (-2, 5, -4) \)
- \( \vec{AC} = C - A = (-2 - 2, -2 - (-3), 3 - 5) = (-4, 1, -2) \)
- \( \vec{AD} = D - A = (3 - 2, 2 - (-3), 4 - 5) = (1, 5, -1) \)

### Step 2: Construct the determinant
The matrix is:
\[
\begin{bmatrix}
-2 & -4 & 1 \\
5 & 1 & 5 \\
-4 & -2 & -1
\end{bmatrix}
\]

### Step 3: Compute the determinant
We expand the determinant as follows:
\[
\text{det} =
-2 \cdot \begin{vmatrix} 1 & 5 \\ -2 & -1 \end{vmatrix}
-4 \cdot \begin{vmatrix} 5 & 5 \\ -4 & -1 \end{vmatrix}
+ 1 \cdot \begin{vmatrix} 5 & 1 \\ -4 & -2 \end{vmatrix}
\]

#### Sub-determinants:
1. \(\begin{vmatrix} 1 & 5 \\ -2 & -1 \end{vmatrix} = (1)(-1) - (5)(-2) = -1 + 10 = 9\)
2. \(\begin{vmatrix} 5 & 5 \\ -4 & -1 \end{vmatrix} = (5)(-1) - (5)(-4) = -5 + 20 = 15\)
3. \(\begin{vmatrix} 5 & 1 \\ -4 & -2 \end{vmatrix} = (5)(-2) - (1)(-4) = -10 + 4 = -6\)

#### Substitute back:
\[
\text{det} = -2(9) - 4(15) + 1(-6) = -18 - 60 - 6 = -84
\]

### Step 4: Calculate the volume
\[
V = \frac{1}{6} \left| -84 \right| = \frac{84}{6} = 14
\]

### Final Answer:
The volume of the tetrahedron is:
\[
\boxed{14 \, \text{units}^3}
\]

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Would you like further clarification or another example?  

Here are five related questions to explore further:
1. How is the determinant formula for tetrahedron volume derived?
2. Can the volume of a tetrahedron be calculated using the cross and dot product?
3. What happens to the volume if one vertex coincides with another?
4. How can the formula be generalized to higher dimensions (e.g., 4D simplex)?
5. What if the vertices were given in parametric form—how would the volume be computed?

### Tip:
When working with determinants, simplify step-by-step to avoid sign errors and double-check sub-determinants.

Вычислить объем тетраэдра с вершинами в точках A(2, -3, 5), B(0, 2, 1), C(-2, -2, 3) и D(3, 2, 4).

Learn how to calculate the volume of a tetrahedron with vertices A(2, -3, 5), B(0, 2, 1), C(-2, -2, 3), and D(3, 2, 4) using determinants and vector operations. This step-by-step solution explains key concepts from analytic geometry and linear algebra.

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