Can lines in 3D be parametrized by the same variable?

  • Context: Undergrad 
  • Thread starter Thread starter inknit
  • Start date Start date
  • Tags Tags
    3d Lines Variable
Click For Summary

Discussion Overview

The discussion revolves around the parametrization of lines in 3D space and whether two lines can be described using the same variable. Participants explore the implications of linear dependence and intersection of the lines based on their parametric equations.

Discussion Character

  • Debate/contested

Main Points Raised

  • Some participants assert that the two lines L1 and L2 do not intersect, suggesting that they can be parametrized independently.
  • Others propose that if the lines are linearly dependent, they could potentially share a parameter, but this is not the case here.
  • A participant mentions that changing the parameter in L1 to "s" leads to an intersection at some point, indicating a need for further analysis.
  • Another participant challenges the idea that the lines cannot intersect due to the use of the same parameter, emphasizing that the parameter's meaning is confined to its respective equation.
  • There is a suggestion to solve the system of equations derived from the parametric forms to determine if the lines intersect.

Areas of Agreement / Disagreement

Participants do not reach a consensus on whether the lines intersect, with some asserting they do not and others suggesting that the parametrization affects the intersection condition. The discussion remains unresolved regarding the intersection of the lines.

Contextual Notes

Participants express uncertainty about the implications of using the same parameter for both lines and the conditions under which they may or may not intersect. There is also a mention of solving a system of equations, but no specific solutions are provided.

inknit
Messages
58
Reaction score
0
For example.

L1: x = 11+3t y = 7+t z = 9+2t

L2: x=-6+4t y=-2+3t z=-7+5t

I was given this problem, and technically these lines don't intersect, right?
 
Physics news on Phys.org
inknit said:
For example.

L1: x = 11+3t y = 7+t z = 9+2t

L2: x=-6+4t y=-2+3t z=-7+5t

I was given this problem, and technically these lines don't intersect, right?

The only way both of these lines can be described by the same variable is if the two lines are linearly dependent.

If this is not the case, you need two variables one for each line.
 
Alright so, they don't intersect correct? B/c if you replace the variable in L1 with let's say "s" they intersect at some point.
 
inknit said:
Alright so, they don't intersect correct? B/c if you replace the variable in L1 with let's say "s" they intersect at some point.

I didn't say they don't intersect. If you want to check what happens solve the linear system L1 = L2.
 
inknit said:
For example.

L1: x = 11+3t y = 7+t z = 9+2t

L2: x=-6+4t y=-2+3t z=-7+5t

I was given this problem, and technically these lines don't intersect, right?
They don't happen to intersect (and not just "technically") but not because the use the same parameter. A parameter has no meaning outside the equation itself. So as not to confuse yourself, it would be better to change one of them to, say, "s". To determine if they intersect, try to solve x= 11+ 3t= -6+ 4s, y= 7+ t= -2+ 3s, z= 9+ 2t= -7+ 5s for s and t.
 

Similar threads

MHB Radius of Sphere Tangent to Two Lines
  • · Replies 2 ·
Replies
2
Views
3K
Evaluate the given integrals - line integrals
  • · Replies 2 ·
Replies
2
Views
1K
Undergrad How Does Parametrization Help Describe Particle Motion in Mathematics?
  • · Replies 3 ·
Replies
3
Views
2K
Prove that PA=2BP in the problem involving parametric equations
  • · Replies 2 ·
Replies
2
Views
1K
Can the contour integral of z⁷ be simplified using a parameterized expression?
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Approximation of a function of two variables
  • · Replies 3 ·
Replies
3
Views
2K
Is (y + z)x1 = (y1 + z1)x the equation of a straight line in 3D space?
  • · Replies 17 ·
Replies
17
Views
3K
MHB Parametric Equations of Line Passing Through (0,1/2,1) and (2,1,-3)
  • · Replies 8 ·
Replies
8
Views
3K
MHB 205.2.6.1-2 another implicit problem with tangent line.
  • · Replies 1 ·
Replies
1
Views
2K
Parametric Equation of Perpendicular Line Through Point of Intersection L1/L2
  • · Replies 3 ·
Replies
3
Views
2K