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To calculate the area of the parallelogram, we can use the formula involving vectors. The area of a parallelogram defined by two vectors \(\vec{u}\) and \(\vec{v}\) is given by the magnitude of their cross product:

\[
\text{Area} = |\vec{u} \times \vec{v}|
\]

1. First, we need to find two vectors that represent the sides of the parallelogram. We can select any two adjacent vertices from the four given points:
   - \((-4,6)\), \((-2,1)\), \((3,6)\), and \((5,1)\).

   Let's pick the vectors between:
   - \( (-4, 6) \) and \( (-2, 1) \)
   - \( (-4, 6) \) and \( (3, 6) \)

2. Compute the vectors:
   \[
   \vec{u} = (-2, 1) - (-4, 6) = (2, -5)
   \]
   \[
   \vec{v} = (3, 6) - (-4, 6) = (7, 0)
   \]

3. Now, calculate the cross product in 2D, which for vectors \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
   \[
   \vec{u} \times \vec{v} = x_1 y_2 - y_1 x_2
   \]

   Substituting the values:
   \[
   \vec{u} \times \vec{v} = (2)(0) - (-5)(7) = 35
   \]

4. The magnitude of the cross product is 35, so the area of the parallelogram is:

\[
\text{Area} = 35
\]

Thus, the area of the parallelogram is 35 square units.

Would you like more details or have any questions?

Here are 5 related questions:
1. How do you compute the area of a parallelogram using determinants?
2. Can we use other pairs of vectors to find the area? Would the result change?
3. How can we calculate the area if the coordinates are in 3D space?
4. How does the cross product help in determining the area?
5. Can we use trigonometry to find the area of a parallelogram?

**Tip:** When using vectors to calculate area, ensure you're selecting adjacent sides of the parallelogram to form the vectors.

Learn how to calculate the area of a parallelogram using vectors with coordinates (-4, 6), (-2, 1), (3, 6), and (5, 1). This problem involves finding vectors, computing their cross product, and determining the area. Suitable for high school students.

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