I have a line $l$ that passes through the point $(3,2,-1)$ and am told that this line intersects the two lines with parametric representations:
$(x,y,z)=(1+t, t, -5+t)$ and $(x,y,z)=(10+5t, 5+t, 2+2t)$
I need to find a parametric representation of the line $l$.
I let the point $(3,2,-1)$ be $O$ and the points of intersection of $l$ and the other two lines be $P(1+t, t, -5+t)$ and $Q(10+5t, 5+t, 2+2t)$, for some real $t$. Then my argument was that if $l$ intersects these two lines, then the points $O,P,Q$ are collinear, which follows that the vectors $OP$ and $PQ$ are parallel. Then I went on to say, for some real $k$, $OP=kPQ$, and tried to solve for $t$, but this did not work. The $t$ I obtained did not produce consistency in $z$-coordinates, only in $x$ and $y$. What is the fault in my argument? What is a better approach?