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To find the area of the triangle \( PQR \) with vertices \( P = (0, -2, -3) \), \( Q = (-2, 2, 1) \), and \( R = (0, 6, 6) \), we can use the formula based on the cross product of two vectors formed by the points. The area \( A \) of the triangle with vertices \( P \), \( Q \), and \( R \) in 3D space is given by:

\[
A = \frac{1}{2} \| \vec{PQ} \times \vec{PR} \|
\]

### Step-by-Step Solution:

1. **Calculate vectors**:
   - \(\vec{PQ} = Q - P = (-2 - 0, 2 - (-2), 1 - (-3)) = (-2, 4, 4)\)
   - \(\vec{PR} = R - P = (0 - 0, 6 - (-2), 6 - (-3)) = (0, 8, 9)\)

2. **Find the cross product \(\vec{PQ} \times \vec{PR}\)**:
   \[
   \vec{PQ} \times \vec{PR} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ -2 & 4 & 4 \\ 0 & 8 & 9 \end{vmatrix}
   \]
   Expanding this determinant:
   \[
   = \mathbf{i} (4 \cdot 9 - 4 \cdot 8) - \mathbf{j} (-2 \cdot 9 - 4 \cdot 0) + \mathbf{k} (-2 \cdot 8 - 4 \cdot 0)
   \]
   \[
   = \mathbf{i} (36 - 32) - \mathbf{j} (-18) + \mathbf{k} (-16)
   \]
   \[
   = 4\mathbf{i} + 18\mathbf{j} - 16\mathbf{k}
   \]
   So, \(\vec{PQ} \times \vec{PR} = (4, 18, -16)\).

3. **Calculate the magnitude of the cross product**:
   \[
   \| \vec{PQ} \times \vec{PR} \| = \sqrt{4^2 + 18^2 + (-16)^2}
   \]
   \[
   = \sqrt{16 + 324 + 256} = \sqrt{596} = 2\sqrt{149}
   \]

4. **Find the area of the triangle**:
   \[
   A = \frac{1}{2} \| \vec{PQ} \times \vec{PR} \| = \frac{1}{2} \times 2\sqrt{149} = \sqrt{149}
   \]

### Final Answer:
\[
\text{Area of } \triangle PQR = \sqrt{149}
\]

Would you like further clarification on any part of this solution? 

Here are five additional questions related to this topic:
1. How do you compute the magnitude of a vector in 3D space?
2. What is the significance of the cross product in vector geometry?
3. How can you verify if three points in space form a triangle or are collinear?
4. How do you determine the orientation of a triangle given three vertices?
5. What are some applications of using the cross product to find areas in physics?

**Tip**: When computing cross products, make sure to follow the correct order of vector entries in the determinant to avoid sign errors.

Find the area of a triangle PQR, where P = (0, -2, -3), Q = (-2, 2, 1), and R = (0, 6, 6).

Find the area of the triangle formed by points P(0, -2, -3), Q(-2, 2, 1), and R(0, 6, 6) in 3D space using vector cross product. This solution involves calculating vectors PQ and PR, finding their cross product, and determining the magnitude to compute the area.

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