Find a vector equation and parametric equations for the line which passes through $(1, 0,6)$ and perpendicular to $x+3y+z=5$.

Question

Statement of the problem:

Find a vector equation and parametric equations for the line which passes through $(1, 0,6)$ and perpendicular to $x+3y+z=5$.

I've gone through Calc. I, II in the last week, and I'm really good at basic real analysis, but I really don't understand this problem. I understand that a plane is determined by a point and a vector which is orthogonal to it. I also notice that if we rewrite the plane given above as $x + 3y + z - 5 = 0$, then a vector orthogonal to this plane is $<1, 3, 1>$.

But what to do with this information, I have no idea. This is the first section I've run into in Stewart where I'm completely baffled by what seems to be a rote memorization of equations.

Answer 1

Since the line is perpendicular to the plane, the normal vector $\langle 1,3,1\rangle$ for the plane is parallel to the line, so the line has vector equation

$\langle x,y,z \rangle=\langle 1,0,6\rangle + t\langle 1,3,1\rangle$

and parametric equations $x=1+t, \;\;y=3t, \;\;z=6+t$.