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To find the area of the parallelogram, we can use the formula:

\[
\text{Area} = \left| \mathbf{u} \times \mathbf{v} \right|
\]

where \(\mathbf{u}\) and \(\mathbf{v}\) are two adjacent vectors that define the parallelogram.

### Step-by-Step Solution:

1. **Identify vertices**: The coordinates of the parallelogram vertices are:
   - \( A(-9, -3) \)
   - \( B(-5, -3) \)
   - \( C(-6, 6) \)
   - \( D(-2, 6) \)

2. **Find vectors**: We can pick two adjacent sides, for example, vectors \(\mathbf{u}\) from \(A\) to \(B\) and \(\mathbf{v}\) from \(A\) to \(C\):
   \[
   \mathbf{u} = B - A = (-5, -3) - (-9, -3) = (4, 0)
   \]
   \[
   \mathbf{v} = C - A = (-6, 6) - (-9, -3) = (3, 9)
   \]

3. **Compute the cross product** (in 2D, this is equivalent to the determinant of the matrix formed by \(\mathbf{u}\) and \(\mathbf{v}\)):
   \[
   \left| \mathbf{u} \times \mathbf{v} \right| = \left| 4 \times 9 - 0 \times 3 \right| = \left| 36 \right| = 36
   \]

Thus, the area of the parallelogram is **36 square units**.

Would you like any further clarification or details?

### Follow-up Questions:
1. How do we compute the area of a parallelogram in 3D space?
2. What is the geometric interpretation of the cross product in vector spaces?
3. Can you explain the determinant's role in calculating the area of a parallelogram?
4. What would happen to the area if one of the vectors was scaled by a factor?
5. How can we verify the parallelogram's shape on a coordinate plane?

#### Tip: 
Always ensure the vectors you choose to calculate the area are adjacent sides, as using non-adjacent sides would lead to incorrect results.

What is the area of the parallelogram given its vertices?

Learn how to calculate the area of a parallelogram given its vertices using vectors and cross products. A step-by-step guide to solving this geometry problem, suitable for students in Grades 9-12.

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