# Lines and planes (Scalar equation + pt. of int.)

## Main Question or Discussion Point

so I've been doing review and I got stuck on two questions, I was wondering if anyone could help me out.
Any help would be appericiated - whether it's a hint on what my first step should be or what's the final answer to confirm whether I'm right or wrong :)

1. Find the scalar equation of the plane that passes through the point (1, 1, 4) and is perpendicular to the line of intersection of the planes x + 2y + z = 1 and 2x + y + 3z = 3.

2. Determine the distance from the point P(2, 3, -1) to the plane 2x + y - 2z + 9 = 0. (I know I can plug the values into the equation, but that doesn't really help me. Could anyone show me a longer way to do this involving more steps? I don't really understand where the equation comes from).

Hurkyl
Staff Emeritus
Gold Member
1. Find the scalar equation of the plane that passes through the point (1, 1, 4) and is perpendicular to the line of intersection of the planes x + 2y + z = 1 and 2x + y + 3z = 3.
Start simple, and build up a solution.

What is the scalar equation equation for a plane?
How do you tell if a plane passes through a point?
What is the scalar equation for a plane passing through that point?
What is the line of intersection of those two planes?
How do you tell if a plane is perpendicular to a line?
What is the scalar equation for a plane perpendicular to that line?
What is the scalar equation for a plane passing through that point and perpendicular to that line?

This is a basic problem solving technique: it's useful in just about anything you do.

2. Determine the distance from the point P(2, 3, -1) to the plane 2x + y - 2z + 9 = 0. (I know I can plug the values into the equation, but that doesn't really help me. Could anyone show me a longer way to do this involving more steps? I don't really understand where the equation comes from).
Starting with definitions is another basic problem solving technique.

What is the definition of the distance from a point to a plane?

HallsofIvy
Homework Helper
Lornt said:
so I've been doing review and I got stuck on two questions, I was wondering if anyone could help me out.
Any help would be appericiated - whether it's a hint on what my first step should be or what's the final answer to confirm whether I'm right or wrong :)

1. Find the scalar equation of the plane that passes through the point (1, 1, 4) and is perpendicular to the line of intersection of the planes x + 2y + z = 1 and 2x + y + 3z = 3.

2. Determine the distance from the point P(2, 3, -1) to the plane 2x + y - 2z + 9 = 0. (I know I can plug the values into the equation, but that doesn't really help me. Could anyone show me a longer way to do this involving more steps? I don't really understand where the equation comes from).
We could better "confirm whether I'm right or wrong" if you would show us what you have done.