Question

determine whether the points are (2, −1, 2), (0, 0, 1), (0, 5, −2), (−1, 3, −1)A. coplanar B. coplanar

Answer 1

Explanation:

the points are not coplanar

lets use the first three points to find the equation of the plane:

we need to find two vectors. we can use vectors from (2,-1,2) to (0,0,1) and from (2,-1,2) to (0,5,-2)

vector 1= (-2,1,-1)

vector 2=(-2,6,-4)

now we can find the normal vector:

n= (-2,1,-1) x (-2,6,-4) = (2,6,8)

so the equation containing first three points is:

2x+6y+8z=0

now lets check if the fourth point, (-1,3,-1), lies on this plane:

2(-1) +6(3) +8(-1) =6

since 6 is not equal to 0, the fourth point does not lie on the plane containing the first three points.

therefore, the points are not coplanar.