To find the area of a parallelogram given its vertices, we can use the formula based on the cross product of two vectors formed by its adjacent sides. The vertices provided are $A(2, 2)$, $B(6, 10)$, $C(8, 5)$, and $D(12, 13)$.

The steps are as follows:

1. Form two vectors representing two adjacent sides of the parallelogram:

   • Vector $\overrightarrow{AB} = B - A = (6 - 2, 10 - 2) = (4, 8)$
   • Vector $\overrightarrow{AC} = C - A = (8 - 2, 5 - 2) = (6, 3)$

2. Compute the cross product of these vectors in two dimensions. The magnitude of the cross product gives the area of the parallelogram: $\text{Area} = | \overrightarrow{AB} \times \overrightarrow{AC} |$ In 2D, the cross product is calculated as: $\overrightarrow{AB} \times \overrightarrow{AC} = |(4 \times 3) - (8 \times 6)| = |12 - 48| = |-36| = 36$

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

Would you like more details, or have any further questions?

Here are some related questions:

1. How do you find the cross product in 3D?
2. What is the difference between a parallelogram and a rectangle?
3. How can you find the perimeter of a parallelogram using vertex coordinates?
4. What is the formula for the area of a triangle using its vertices?
5. How do vector operations relate to geometric properties like area?

Tip: The cross product is a great way to find areas of parallelograms and triangles in vector spaces!