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To find the volume of a tetrahedron given vertices \( A(5, 4, -3) \), \( B(-6, -6, -5) \), \( C(4, -6, -4) \), and \( D(0, 0, 10) \), we can use the following formula:

\[
V = \frac{1}{6} \left| \vec{AB} \cdot (\vec{AC} \times \vec{AD}) \right|
\]

### Step-by-Step Solution

1. **Find Vectors \( \vec{AB} \), \( \vec{AC} \), and \( \vec{AD} \):**

   \[
   \vec{AB} = (-6 - 5, -6 - 4, -5 + 3) = (-11, -10, -2)
   \]
   \[
   \vec{AC} = (4 - 5, -6 - 4, -4 + 3) = (-1, -10, -1)
   \]
   \[
   \vec{AD} = (0 - 5, 0 - 4, 10 + 3) = (-5, -4, 13)
   \]

2. **Compute the Cross Product \( \vec{AC} \times \vec{AD} \):**

   Using the determinant for the cross product:
   \[
   \vec{AC} \times \vec{AD} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ -1 & -10 & -1 \\ -5 & -4 & 13 \end{vmatrix}
   \]

   Expanding the determinant:
   \[
   \vec{AC} \times \vec{AD} = \hat{i}( (-10)(13) - (-1)(-4) ) - \hat{j}( (-1)(13) - (-1)(-5) ) + \hat{k}( (-1)(-4) - (-10)(-5) )
   \]

   Simplifying each component:
   \[
   = \hat{i}(-130 - 4) - \hat{j}(-13 - 5) + \hat{k}(4 - 50)
   \]
   \[
   = \hat{i}(-134) + \hat{j}(18) - \hat{k}(46)
   \]
   So,
   \[
   \vec{AC} \times \vec{AD} = (-134, 18, -46)
   \]

3. **Calculate the Dot Product \( \vec{AB} \cdot (\vec{AC} \times \vec{AD}) \):**

   \[
   \vec{AB} \cdot (\vec{AC} \times \vec{AD}) = (-11)(-134) + (-10)(18) + (-2)(-46)
   \]
   \[
   = 1474 - 180 + 92 = 1386
   \]

4. **Calculate the Volume:**

   \[
   V = \frac{1}{6} |1386| = \frac{1386}{6} = 231
   \]

### Final Answer

The volume of the tetrahedron is:
\[
\boxed{231}
\]

Would you like further details or have any questions?

---

1. How is the cross product of two vectors calculated in 3D?
2. What are some other methods to find the volume of a tetrahedron in 3D?
3. How does the determinant help in calculating vector products?
4. Why is the factor \( \frac{1}{6} \) used in the volume formula for a tetrahedron?
5. Can this method be applied to any shape other than a tetrahedron?

**Tip:** Always double-check calculations, especially signs, when dealing with vector products, as mistakes can lead to incorrect volume values.

Find the volume of the tetrahedron having the given vertices: (5, 4, -3), (-6, -6, -5), (4, -6, -4), (0, 0, 10).

Learn how to calculate the volume of a tetrahedron with vertices at (5, 4, -3), (-6, -6, -5), (4, -6, -4), and (0, 0, 10) using vector geometry. The solution involves cross product, dot product, and vector algebra. Suitable for high school students in grades 11-12.

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