Finding the Equation of a Plane Perpendicular to a Given Line and Plane

[BraedenP]

Homework Statement

Find the equation of the plane that contains the line $x=-1+3t, y=5+2t, z=2-t$ and is perpendicular to the plane $2x-4y+2z=9$

Homework Equations

Equation of a plane:
$$a(x-x_o)+b(y-y_o)+c(z-z_o)=0$$

The Attempt at a Solution

I was thinking that I could simply find an orthogonal normal to the normal of the specified plane. That would give me the normal of my new plane.

I found such a normal to be (3, 1, -1).

Then I'd simply take a the direction of the line (3, 2, -1), and plop it into the plane equation. This is where I get stuck; I can't figure out how to get the point specification into that equation. Is it simply:

$$3(3)x+1(2)y-1(-1)z = 9x+2y+z = 0$$

It seems too easy for me. What am I doing wrong?

 

[Mark44]

Homework Statement

Find the equation of the plane that contains the line $x=-1+3t, y=5+2t, z=2-t$ and is perpendicular to the plane $2x-4y+2z=9$

Homework Equations

Equation of a plane:
$$a(x-x_o)+b(y-y_o)+c(z-z_o)=0$$

The Attempt at a Solution

I was thinking that I could simply find an orthogonal normal to the normal of the specified plane. That would give me the normal of my new plane.

I found such a normal to be (3, 1, -1).

This is one vector that is perpendicular to the normal of the plane you are to find. The problem is that there are an infinite number of vectors that are perpendicular to that plane.

Then I'd simply take a the direction of the line (3, 2, -1), and plop it into the plane equation. This is where I get stuck; I can't figure out how to get the point specification into that equation. Is it simply:

$$3(3)x+1(2)y-1(-1)z = 9x+2y+z = 0$$

It seems too easy for me. What am I doing wrong?

 

[BraedenP]

This is one vector that is perpendicular to the normal of the plane you are to find. The problem is that there are an infinite number of vectors that are perpendicular to that plane.

There are, but all the rest of them would simply be scalar multiples of that one, right? Therefore, it doesn't matter which multiple is used in the equation; it'll work either way.

 

[Mark44]

No, not at all. All of the vectors that are perpendicular to <2, -4, 2> would lie in the same plane, but they point in all different directions.