Solve 3D Plane Intersection & Reflection: Steps Included!

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Discussion Overview

The discussion revolves around the intersection of a line and a plane in three-dimensional space, specifically focusing on determining the intersection point and the equation for the reflection of the line in the plane. The context includes both theoretical and practical aspects relevant to a unit test on 3D planes.

Discussion Character

  • Homework-related
  • Mathematical reasoning

Main Points Raised

  • One participant inquires about the intersection of a line and a plane, seeking steps to determine the reflection of the line in the plane.
  • Another participant asserts that a line and a plane intersect at a single point and provides a specific example with equations for both the plane and the line, outlining the substitution method to find the intersection point.
  • A third participant references an external resource that explains how to find the reflection of the line in the plane.
  • A later reply expresses gratitude for the information provided, indicating it was helpful.

Areas of Agreement / Disagreement

Participants generally agree on the intersection of a line and a plane occurring at a single point, but the discussion does not resolve the specifics of the reflection process, as it relies on an external resource.

Contextual Notes

The discussion does not clarify the assumptions or definitions regarding the reflection process, nor does it address any potential limitations in the provided example.

Saad
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I have a unit test on 3D-Planes coming up! And I was wondering if in 3D, a line and a plane intersect at a single point. How would you determine the equation of the reflection of the line in the plane?
Please try to provide the steps. Thanks!
 
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Yes, a plane and a line intersect in one point. Let's say you have the plane:
[tex]\pi : 7x + 8y + 9z + 10 = 0[/tex]
And the line:
[tex]\underline{l} = (1, 2, 3) + t(4, 5, 6)[/tex]
You know that the X, Y and Z coordinates of every point on the line are:
[tex]x = 1 + 4t[/tex]
[tex]y = 2 + 5t[/tex]
[tex]z = 3 + 6t[/tex]
Substitute these in the equation of the plane and you will get:
[tex]7(1 + 4t) + 8(2 + 5t) + 9(3 + 6t) + 10 = 0[/tex]
One unknown - t. Find it and you can find the coordinates of the intersection point.
 
really appreciate it, this was helpful! thanks!
 

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