Thetwovectors 𝑢⎯⎯=⟨−1,2,2⟩ u ¯ = ⟨−1 ,2 , 2⟩ and 𝑣⎯-=⟨−3,−3,2⟩ v ¯ = ⟨ − 3 , − 3 , 2 ⟩ determine a plane in space. Mark vectors below asT if the vector lies in same plane as 𝑢⎯⎯⎯u ¯ and 𝑣⎯⎯⎯ v ¯

Question

1. <2,2,3>
2. <9,9,-6>
3. <-1,2,0>
4. <-5,-8,2>

Answer 1

Calculate the cross-product of u and v. That defines the equation of the plane. Then, check to see if each of 1-4 is on the plane.

I will show the cross-product and 4, you can then do 1-3

i j k

-1 2 2

-3 -3 2

10i - 4j + 9k

the equation of the plane is

10x - 4y + 9z = 0

For (4), (-5, -8, 2)

10(-5) -4(-8) + 9(2) = -50 + 32 + 18 = 0

(-5, -8, 2) is on the plane.