We didn't go over how to solve this problem in my class, and I'm trying to piece it together from our textbook and online sources, but I'm still not getting anything.
The question is: Find a parametrization for the line perpendicular to $(2, −1, 1)$, parallel to the plane $2x + y − 4z = 1$, and passing through the point $(1, 0, −3)$.
What I tried doing was saying that the directional vector of $l(t)$ would be perpendicular to $(2, -1, 1)$, so their dot product should equal zero. From this I got $2x-y+z=0$.
I then plugged the point given into $2x + y − 4z = c$, to find the plane that this new line would be on, and got $2x+y-4z=14$.
We were taught to arbitrarily set one of these values $(x, y, z)$ equal to zero, so I chose to do that for z. Then, solving for the remaining variables, I get $x=7/2$ and $y=0$.
Therefore, my equation for the line $l(t)=(1+7t/2, 0, -3)$, but the online homework system is not accepting this. Any help to point out where I went wrong and explain this problem would be much appreciated!
Best Answer
HINT: The direction vector of your line should be perpendicular to both $(2,-1,1)$ and $(2,1,-4)$. So in what direction should it lie?