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

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Discussion Overview

The discussion revolves around two mathematical problems related to planes and distances in three-dimensional space. The first problem involves finding the scalar equation of a plane that passes through a specific point and is perpendicular to the line of intersection of two given planes. The second problem concerns determining the distance from a point to a specified plane, with a request for a more detailed explanation of the process involved.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Technical explanation

Main Points Raised

  • One participant seeks help with finding the scalar equation of a plane that passes through the point (1, 1, 4) and is perpendicular to the line of intersection of the planes defined by the equations x + 2y + z = 1 and 2x + y + 3z = 3.
  • Another participant suggests starting with basic definitions and problem-solving techniques to approach the first question, asking about the scalar equation for a plane and how to determine if it passes through a point.
  • The same participant also inquires about the definition of the distance from a point to a plane in relation to the second question, emphasizing the need for a more detailed step-by-step explanation.
  • A later reply references an external link that may provide solutions related to the problems discussed, although it does not directly address the questions posed.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus on the solutions to the problems presented. The discussion remains open, with multiple approaches and requests for clarification without definitive answers provided.

Contextual Notes

Some assumptions regarding the definitions of planes and distances may be missing, and the discussion does not resolve the mathematical steps involved in the problems. There is also a lack of clarity on the specific methods to be used for solving the problems.

Lornt
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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).
 
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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?
 
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.
 

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