I'm doing some practice problems and have the following question which I am able to produce a 'correct' result for however I believe my reasoning is off.
Question: Show that $L_1$ and $L_2$ are equations of the same line.
$L_1: (x,y,z)=(3,1,-7)+\lambda(2,2,-1), \qquad L_2: 2(6-x)=2(4-y)=2(2z+17)$
Rewrite $L_2$ as $2x+2y+8z=-48$.
I can then say $(2,2,-1)$ is parallel to $L_1$ and $(2,2,8)$ is perpendicular to $L_2$. Equate the dot product to zero, showing the lines are parallel, then input $(3,1,-7)$ into my rewritten $L_2$ to show the point on $L_1$ also falls on $L_2$, hence they are the same line.
Is this an incorrect approach? My issue is that my rewritten $L_2$ appears to be an equation for a plane which $L_2$ lies completely within, and then I'm selecting the only vector which is perpendicular to the plane, even though it would be one of infinite vectors perpendicular to the line. Or would any plane that $L_2$ lies completely within have a normal vector perpendicular to $L_2$ which covers my concern?