Intersection/Collision of two lines in R^3

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To determine if the lines r1 and r2 intersect or collide, one should set the corresponding components of the parametric equations equal to each other, resulting in three equations to solve for t1 and t2. A collision occurs when both lines intersect at the same point in space, meaning t1 and t2 yield the same values and satisfy all three equations simultaneously. The discussion clarifies that while all collisions are intersections, not all intersections are collisions, as there may be points where the lines intersect at different parameter values. The approach to solving the problem remains consistent with previous similar problems. Understanding the distinction between collision and intersection is crucial for accurately analyzing the lines in R^3.
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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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