Find the area of the parallelogram with vertices A=(3,2,0),B=(8,3,0),C=(5,7,,0),and D=(10,8,0).

The area of a parallelogram is given by the cross product of two of its sides.

Step 1: Find the vectors representing two sides of the parallelogram. We can use AB and AD as the two sides.

Vector AB = B - A = (8-3, 3-2, 0-0) = (5, 1, 0) Vector AD = D - A = (10-3, 8-2, 0-0) = (7, 6, 0)

Step 2: Find the cross product of the two vectors. The cross product of two vectors (a, b, c) and (d, e, f) is given by (bf - ce, cd - af, ae - bd).

So, the cross product of AB and AD is:

= (10 - 06, 07 - 50, 56 - 17) = (0, 0, 30 - 7) = (0, 0, 23)

Step 3: Find the magnitude of the cross product. The magnitude of a vector (a, b, c) is given by sqrt(a^2 + b^2 + c^2).

So, the magnitude of the cross product is:

= sqrt(0^2 + 0^2 + 23^2) = sqrt(0 + 0 + 529) = sqrt(529) = 23

So, the area of the parallelogram is 23 square units.