# Intersection/Collision of two lines in R^3

In summary, the conversation discusses how to determine if two lines, r1 and r2, intersect or collide using a system of equations and solving for the values of t. The approach for determining intersect is the same for collision, but in collision, the lines intersect at the same values of t for all equations. The specific example provided shows that if t=3 satisfies all equations for x, y, and z, then it is a collision.

## 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

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?