How can we determine the intersection point of two lines with given equations?

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Discussion Overview

The discussion revolves around determining the relationship between two lines given in symmetric form, specifically whether they are parallel, skew, or intersecting, and if they intersect, finding the point of intersection. The scope includes mathematical reasoning and technical explanations related to vector equations and parametric forms.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • Some participants propose that lines are parallel if their direction vectors are scalar multiples of each other.
  • Others argue that two lines intersect if there exists a common point where their coordinates match.
  • A participant notes that skew lines do not intersect and are not parallel.
  • There is a discussion about using parametric equations to find intersection points by equating the x, y, and z coordinates.
  • One participant expresses confusion about the term "equating" and the process of solving for parameters.
  • Participants discuss the implications of finding parameters s and t that satisfy the equations, indicating whether the lines intersect or are skewed.
  • There is a correction regarding the equations presented, with a participant realizing a mistake in the symmetric form of one of the lines.
  • Participants explore the method of elimination to solve the equations for parameters s and t, with some expressing uncertainty about the process.

Areas of Agreement / Disagreement

Participants generally agree on the definitions of parallel, intersecting, and skew lines, but there is ongoing discussion regarding the specific calculations and methods to determine the intersection point. The discussion remains unresolved regarding the exact parameters and whether the lines intersect.

Contextual Notes

Participants express uncertainty about the correct forms of the equations and the process of solving them, indicating potential limitations in their understanding of the mathematical concepts involved.

Who May Find This Useful

This discussion may be useful for students or individuals interested in understanding the relationships between lines in three-dimensional space, particularly in the context of vector mathematics and geometry.

  • #31
Yep, now apply the methods of checking I mentioned above. (Nod)
 
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  • #32
im actually not getting the same answers. i think i made a mistake.
 
  • #33
t=2
 
  • #34
it does satisfy the third equation as well. so now i plug in s and t into the parametric equations and and get x,y,z which turns out to be (4,-1,-5) and i checked with the back of the book. it's right :)
 
Last edited:

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