Question

Find the point(s) of intersection (if any) of the plane and the line. Also, determine whether the line lies in the plane. 2x - 2y + z = 12, x - 1/2 = y + (3/2)/-1 = (z + 1) / 2

Answer 1

Answer:

the intersection is the point P=(x,y,z)=(8,9,14) and the line does not lie in the plane

Step-by-step explanation:

from the equation of the line

x - 1/2 = y + (3/2)/-1 = (z + 1) / 2  = t (parameter)

then the parametric equation of the line is

x= 1/2 +t

y =  (3/2) + t

z = (-1) + 2*t

therefore from the equation of the plane

2x - 2y + z = 12

2*(1/2 +t) - 2[(3/2) + t]  + [(-1) + 2*t]  = 12

1+2*t - 3 -2*t -1 + 2*t = 12

-3 + 2*t = 12

t= 15/2

therefore there is only one intersection of the line with the plane ( then the line does not lie in the plane , since there would be infinite intersection points). The intersection is

x= 1/2 +t = 1/2 +15/2 = 8

y =  (3/2) + t = (3/2) + 15/2 = 9

z = (-1) + 2*t =  (-1) + 2*15/2  = 14

thus the intersection point P=(x,y,z)=(8,9,14)