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To calculate the area of the parallelogram, we can use the **cross product** of two vectors that define adjacent sides of the parallelogram. Given the vertices of the parallelogram, we can form two vectors.

### Step 1: Identify the coordinates of the vertices
The vertices of the parallelogram are:
- \( A(-4, 6) \)
- \( B(3, 6) \)
- \( C(5, 1) \)
- \( D(-2, 1) \)

We can now define two vectors using adjacent vertices, say:
- Vector \( \vec{AB} \)
- Vector \( \vec{AD} \)

### Step 2: Calculate the vectors
- \( \vec{AB} = B - A = (3 - (-4), 6 - 6) = (7, 0) \)
- \( \vec{AD} = D - A = (-2 - (-4), 1 - 6) = (2, -5) \)

### Step 3: Find the magnitude of the cross product
For vectors \( \vec{v} = (x_1, y_1) \) and \( \vec{w} = (x_2, y_2) \), the cross product in two dimensions is given by:
\[
|\vec{v} \times \vec{w}| = |x_1 y_2 - x_2 y_1|
\]

Substituting the values of \( \vec{AB} = (7, 0) \) and \( \vec{AD} = (2, -5) \):
\[
|\vec{AB} \times \vec{AD}| = |7 \cdot (-5) - 2 \cdot 0| = |-35| = 35
\]

### Step 4: Conclusion
The area of the parallelogram is 35 square units.

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Do you want more details on the cross product method, or do you have any questions?

Here are 5 related questions you may find helpful:
1. How do you calculate the area of a triangle using vectors?
2. What are some other methods to find the area of a parallelogram?
3. How is the magnitude of a vector defined in 2D space?
4. What is the geometric meaning of the cross product of two vectors?
5. How do you compute the perimeter of the parallelogram from its vertices?

**Tip:** When dealing with vectors in geometry, the cross product is useful for determining areas of parallelograms or triangles, as it gives the perpendicular area between two vectors.

Learn how to calculate the area of a parallelogram using vectors in 2D. Given the vertices A(-4, 6), B(3, 6), C(5, 1), and D(-2, 1), this solution explains how to form vectors AB and AD, and use their cross product to find the area. Suitable for Grades 9-12.

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