find the area of the parallelogram that has the vectors as adjacent sides
u = -2i + j + 5k
v = 4i - 3j - 3k
a. 11
b. squareroot 26
c. 86
d. 2 squareroot 86
e. 1
the area is |u×v| = |12i+14j+2k|
To find the area of a parallelogram given the vectors as adjacent sides, you can use the cross product of the two vectors. The cross product of two vectors gives another vector that is perpendicular to both of the original vectors. The magnitude of this cross product vector is equal to the area of the parallelogram formed by the original two vectors.
First, calculate the cross product of vectors u and v:
u x v = (4 * (5) - (-3) * (1))i - ((-2) * (5) - (-3) * (-2))j + ((-2) * (1) - 4 * (-3))k
= (20 + 3)i - (10 + 6)j - (1 - 12)k
= 23i - 16j + 11k
Next, calculate the magnitude of the cross product vector:
|u x v| = sqrt((23)^2 + (-16)^2 + 11^2)
= sqrt(529 + 256 + 121)
= sqrt(906)
= sqrt(6 * 151)
= sqrt(6) * sqrt(151)
Therefore, the area of the parallelogram is sqrt(6) * sqrt(151), which simplifies to 2 sqrt(151) (option d).