The area of a parallelogram with given vertices in rectangular coordinates can be calculated using the vector cross product. The area of a parallelogram is equal to the product of its base and height. Using vector values derived from the vertices, the product of a parallelogram's base and height is equal to the cross product of two of its adjacent sides. Calculate the area of a parallelogram by finding the vector values of its sides and evaluating the cross product.

Find the vector values of two adjacent sides of the parallelogram by subtracting the x and y values of the two vertices that form the side. For example, to find length DC of parallelogram ABCD with vertices A (0, -1), B (3, 0), C (5, 2) and D (2, 1), subtract (2, 1) from (5, 2) to get (5 - 2, 2 - 1) or (3, 1). To find length AD, subtract (2, 1) from (0, -1) to get (-2, -2).

Write a matrix of two rows by three columns. Fill in the first row with the vector values of one side of the parallelogram (the x value in the first column and the y value in the second) and write zero in the third column. Fill in the values of the second row with the vector values of the other side and zero in the third column. In the above example, write a matrix with the values {{3 1 0}, {-2 -2 0}}.

Find the x-value of the cross product of the two vectors by blocking out the first column of the 2 x 3 matrix and calculating the determinant of the resulting 2 x 2 matrix. The determinant of a 2 x 2 matrix {{a b}, {c d}} is equal to ad - bc. In the above example, the x-value of the cross product is the determinant of the matrix {{1 0}, {-2 0}}, which is equal to 0.

Find the y-value and z-value of the cross product by blocking out the second and third columns of the matrix, respectively, and calculating the determinant of the resulting 2 x 2 matrices. The y-value of the cross product is equal to the determinant of the matrix {{3 0}, {-2 0}}, which is equal to zero. The z-value of the cross product is equal to the determinant of the matrix {{3 1}, {-2 -2}}, which is equal to -4.

Find the area of the parallelogram by calculating the magnitude of the cross product