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.