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)
