Intersection/Collision of two lines in R^3

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Homework Help Overview

The problem involves determining whether two parametric lines in three-dimensional space intersect or collide, defined by the equations r1(t) and r2(t). The original poster expresses confusion regarding the concepts of intersection and collision, seeking clarification on how to approach the problem.

Discussion Character

  • Exploratory, Conceptual clarification, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the method of setting the components of the two lines equal to each other to find potential intersection points. The original poster reflects on a previous similar problem and questions the meaning of 'collision' in this context. Some participants suggest that 'collision' implies the lines intersect at the same parameter value for both equations.

Discussion Status

The discussion is ongoing, with participants exploring the definitions of intersection and collision. Some guidance has been provided regarding the approach to solving the problem, but there is no explicit consensus on the definitions or the implications of the findings.

Contextual Notes

The original poster mentions feeling lost and requests hints rather than complete solutions. There is a focus on understanding the terminology and the implications of the results rather than on solving the equations directly.

pearss
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Homework Statement



Determine whether r1 and r2 collide or intersect:

r1(t) = <t^2 + 3 , t + 1 , 6t^-1 >

r2(t) = <4t , 2t -2 , t^2 - 7>

I am completely lost in this problem and was hoping for a just a hint at where to begin. I'm unsure what it even means if two lines collide or intersect.

I've done a similar problem that read:

Determine if

r1(t) = < 1 , 0 , 1 > + t<3, 3, 5 >

and

r2(t) = < 3, 6, 1 > +t<4, -2, 7>

intersect.

I did it by multiplying the scalars out and adding the two vectors. Then setting the x components of the two lines equal to each other...same with y and z. This gives me three equations with which i use to solve for t1 and t2. Finally, plugging the t values into the third equation will prove whether or not the lines intersect if the equation is satisfied with the two t values.

I'm unsure what 'collision' is. Do i approach this problem the same way?

Thanks all
 
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Yes, do it the same way for intersect. 'Collide' I think means that they intersect with the same value of t in each equation. I.e. they are at the same place at the same time.
 
so if i find the value for t1 to be 3 and the value for t2 to be 3 and they satisfy all equaitons for x, y and z then these lines collide because both "t" values are the same and all intersections are the same?
 
pearss said:
so if i find the value for t1 to be 3 and the value for t2 to be 3 and they satisfy all equaitons for x, y and z then these lines collide because both "t" values are the same and all intersections are the same?

Yes, t=3 is a collision. There MIGHT be more intersections that aren't collisions. But in this case I don't think there are.
 
pearss said:
so if i find the value for t1 to be 3 and the value for t2 to be 3 and they satisfy all equaitons for x, y and z then these lines collide because both "t" values are the same and all intersections are the same?

Yes, t=3 is a collision. There MIGHT be more intersections that aren't collisions. But in this case I don't think there are.
 

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