Answer
$7x+y+9z=25$
Work Step by Step
$x=-2+t$ $y=3+2t$ $z=4-t$
Vector parallel to line is $<1,2,-1>$
$t=0$ implies $(-2,3,4)$ lies on the line. (Point P)
$t=1$ implies $(-1,5,3)$ lies on the line. (Point Q)
$x=3-t$ $y=4-2t$ $z=t$
Vector parallel to line is $<-1,-2,1>$
$t=0$ implies $(3,4,0)$ lies on the line. (Point R)
$<1,2,-1>=(-1)<-1,-2,1>$
Thus the lines are parallel.
$\vec{PQ}=<-1,5,3>-<-2,3,4,>=<1,2,-1>$
$\vec{PR}=<3,4,0>-<-2,3,4>=<5,1,-4>$
$\vec{PQ}\times\vec{PR}=<-7,-1,-9>$ is the normal to the plane. Point P lies on the plane. Thus the equation of the plane is:
$[-<-2,3,4>]\cdot<-7,-1,-9>=0$
$\Longrightarrow \cdot<-7,-1,-9>=0$
$\Longrightarrow -7(x+2)-(y-3)-9(z-4)=0$
$\Longrightarrow 7(x+2)+(y-3)+9(z-4)=0$
$7x+y+9z=25$ is the equation of the plane.
