Intersection/Collision of two lines in R^3

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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.
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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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  • #2
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.
 
  • #3
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?
 
  • #4
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.
 
  • #5
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.
 

1. What is the formula for finding the intersection point of two lines in R^3?

The formula for finding the intersection point of two lines in R^3 is called the parametric form and is given by:
x = x1 + at,
y = y1 + bt,
z = z1 + ct
where (x1, y1, z1) are the coordinates of a point on the first line, and a, b, and c are the direction numbers of the first line. t is a parameter that represents the distance along the line.

2. How many possible outcomes can occur when two lines intersect in R^3?

There are three possible outcomes when two lines intersect in R^3:
1. The lines intersect at a single point
2. The lines are coincident (they lie on top of each other)
3. The lines are parallel (they do not intersect)
The outcome depends on the relationship between the direction numbers of the two lines.

3. Can two lines intersect at more than one point in R^3?

No, two lines in R^3 can only intersect at one point. This is because three-dimensional space is uniquely defined by three coordinates, so two lines cannot occupy the same point in space simultaneously.

4. How can I determine if two lines are parallel or coincident in R^3?

To determine if two lines are parallel or coincident, you can compare their direction numbers. If the direction numbers of the two lines are proportional to each other, then the lines are parallel. If the direction numbers are equal, then the lines are coincident.

5. Is there a way to find the intersection point of two lines in R^3 without using the parametric form?

Yes, there is another method for finding the intersection point of two lines in R^3 called the symmetric form. This involves setting the x, y, and z equations of the two lines equal to each other and solving for the values of t. However, this method only works for lines that are not parallel or coincident.

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