I need to find the equation of a line $s$ which is perpendicular to the plane $\pi : x-2y+z=35$ and the line is passing through $P(1, 2, 3)$.
To get the parametric equation of $s$, since the line is perpendicular to the plane, the plane and the line have the same direction so I take the coefficients of $xyz$ from the plane equation, getting
$s: \begin{cases} x=1+t\\y=2-2t\\z=3+t \end{cases}$
Now, I need to find the equation of a line $r$ too. I know that it's passing through $Q(1, 2, 3)$ and $R(3, 2, a)$.
Its equation is $ r: \begin{cases} x=1+2t \\y=2\\z=3+(a-3)t \end{cases}$
I now need to find if the lines are parallel, and if they are, for which values of $a$.
I get the two directional vectors of the two lines:
$ v=(1, -2, 1) $
$w=(2, 0, a-3)$
For the lines to be parallel, the directional vectors should be linearly dependent, right? However, they'll always be LI regardless of the value of $a$. So am I right in saying that the two lines will never be parallel for any value of $a$?
The two lines will indeed never be parallel. They intersect in $P=Q$. Moreover, $R$ is on line $r$, but not on line $s$ whatever the value of $a$.