Does the line intersect the plane [x,y,z] = [4, -15, -8] + s[1,-3,1] + t[2,3,1]?
a)[x,y,x] = [5,-9, 3] + k[1, -12, 2]
I know you are supposed to find the dot product of the normal and the direction vector, but i'm used to the planar equation being in ax+by+cz+d=0 form and i don't know how to get the normal from this one.
Any help would be greatly appreciated :D
Hint:
Since the vectors $<1,-3,1>$ and $<2,3,1>$ are parallel to the plane, you can take their cross-product to get a normal vector $<-6,1,9>$ for the plane.
An alternate way to do this would be to solve the system of equations
$x=4+s+2t$, $y=-15-3s+3t$, $z=-8+s+t$
by, say, eliminating $s$ from the first two equations and then from the first and third equations, and then eliminating $t$ from the resulting equations.