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To find the area of a parallelogram given its vertices, we can use the vector cross product. Here's a step-by-step approach:

1. **Vertices provided**:
   - \( A(-2, 1) \)
   - \( B(0, 4) \)
   - \( C(4, 2) \)
   - \( D(2, -1) \)

2. We know that the area of a parallelogram can be found using the formula:
   \[
   \text{Area} = \left| \mathbf{u} \times \mathbf{v} \right|
   \]
   where \( \mathbf{u} \) and \( \mathbf{v} \) are vectors formed by the sides of the parallelogram. 

3. **Find two vectors** that represent adjacent sides of the parallelogram:
   - Vector \( \mathbf{AB} = B - A = (0 - (-2), 4 - 1) = (2, 3) \)
   - Vector \( \mathbf{AD} = D - A = (2 - (-2), -1 - 1) = (4, -2) \)

4. **Calculate the cross product** in two dimensions, which in 2D is simply:
   \[
   \mathbf{u} \times \mathbf{v} = | \mathbf{AB} \times \mathbf{AD} | = \left| (2)(-2) - (3)(4) \right| = | -4 - 12 | = |-16| = 16
   \]

Thus, the area of the parallelogram is **16 square units**.

Would you like further details or clarification?

Here are 5 related questions you might find interesting:
1. How do you compute the area of a parallelogram in 3D space?
2. What happens to the area of a parallelogram when all vertices are shifted by the same vector?
3. How can you use determinants to find the area of a parallelogram?
4. How do you find the area of a triangle using a similar method?
5. How can you generalize this method to find the area of polygons?

**Tip**: When working with polygons, breaking them down into smaller geometric shapes, like triangles or parallelograms, can simplify area calculations.

Find the area of the parallelogram with vertices A(-2, 1), B(0, 4), C(4, 2), and D(2, -1).

Learn how to calculate the area of a parallelogram with vertices A(-2, 1), B(0, 4), C(4, 2), and D(2, -1) using vector cross product. This problem covers vector geometry concepts and cross product calculations. Suitable for high school students.

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