Without solving , determine the points of intersection of the line r= (5,-9,3) +k (1,-12,2] and the plane [x,y,z]= (4,-15,-8) + s[1,-3,1] + t[2,3,1], if any exist.
line r= (5,-9,3) +k (1,-12,2]
x = 5 + k
y = -9 - 12k
z = 3 + 2k
plane:
x = 4 + s + t
y = -15 - 3s + 3t
z = -8 + s + t
so
4 + s+ t = 5 + k
s + t - k = 1 , #1
-15 - 3s + 3t = -9 - 12k
-3s + 3t + 12k = 6 , #2
-8 + s + t = 3 + 2k
s + t - 2k = 11 , #3
#1 - #3:
k = -10
sub back into line equation:
r = (5,-9,3) - 10(1,-12,2)
= (-5 , 111 , -17)