Final answer:
To find the area of a parallelogram with two adjacent sides u and v, calculate the cross product of u and v and find the magnitude of the cross product.
Step-by-step explanation:
To find the area of the parallelogram with two adjacent sides u and y, we can use the cross product of the vectors u and v. First, calculate the cross product using the formula u x v = |u| |v| sin(theta) n, where |u| and |v| are the magnitudes of u and v, theta is the angle between u and v, and n is the unit vector perpendicular to both u and v.
Once you have the cross product, calculate its magnitude to find the area of the parallelogram.
Let's calculate the cross product:
u x v = (3i - j) x (3j + 2k) = (3 * 2)i + (3 * 3)j + ((-1 * 3) - (2 * 3))k = 6i + 9j - 9k
Now, calculate the magnitude of the cross product:
|u x v| = sqrt((6)^2 + (9)^2 + (-9)^2) = sqrt(36 + 81 + 81) = sqrt(198)
Therefore, the area of the parallelogram is sqrt(198) square units.