Final answer:
To find the area of a parallelogram with two adjacent sides, u and v, you can use the cross product of the two sides. The cross product of u and v is 4j + 7k. The magnitude of this vector, |u × v|, is equal to the area of the parallelogram.
Step-by-step explanation:
To find the area of a parallelogram, A, with two adjacent sides, u and v, you can use the cross product of the two sides. The cross product of two vectors gives a new vector that is perpendicular to both original vectors, whose magnitude is equal to the area of the parallelogram.
The formula for finding the cross product of two vectors is u × v = (u.y*v.z - u.z*v.y)i - (u.x*v.z - u.z*v.x)j + (u.x*v.y - u.y*v.x)k.
Given that u = 2i - j - 2k and v = 3i + 2j - k, we can substitute the values into the formula and perform the cross product to find the area:
u × v = ((-2)(-1)-(2)(2))i - ((2)(-1)-(3)(2))j + ((2)(2)-(3)(-1))k
u × v = 0i - (-4j) + (7k)
u × v = 4j + 7k
So, the cross product u × v is the vector 4j + 7k. The magnitude of this vector, |u × v|, is equal to the area of the parallelogram. Therefore, the area of the parallelogram with adjacent sides u and v is 4.