Discovering Parallel, Intersecting, and Skew Lines: A Vector Calculus Problem

[lijet13]

Ok here is the problem:

Given two lines in space, either they are parallel, or they intersect or they are skew. Determine whether the lines taken two at a time, are parallel, intersect or are skew. If they intersect find the point of intersection.

line 1: x=1+2t, y=-1-t, z=3t; -infiniti<t<infiniti
line 2: x=2-s, y=3s, z=1+2; -infiniti<s<infiniti
line 3: x=5+2r, y=1-r, z=8+3r; -infiniti<r<infiniti

I'm not really sure where to go with this. I found the normal vector forms of the equations and I don't think any of them are parallel using the cross product=0 when vectors are parallel but i have no idea how to find itnersection or skew when given the parametric equations. Did I do the parallel part right? and where would I begin for the other parts. Do you use each part (x,y,z) from the parametric as three points to find the equation of the plane formed?

Thanks so much for any help

 

[StatusX]

Pick some axis, say z, (make sure none of the lines are perpendicular to this axis first; I haven't checked) and solve for x and y in terms of z for each line. Then solve for the z where two lines have the same x, say. There are a few possibilities:

1. This equation is inconsistent (reduces to somethine like 1=2).
2. The equation holds for all z (reduces to something like 1=1).
3. The y values are also the same at this z.
4. The y values are different at this z.

I'll let you figure out which each case means, but as a clue, I'll tell you that 2 of the above cases tell you they don't intersect, one tells you they do and one requires a little more work to get an answer.

 

[quicknote]

!

I would approach this problem by first understanding the definitions of parallel, intersecting, and skew lines. Parallel lines are lines that never intersect and have the same slope. Intersecting lines are lines that cross at a single point. Skew lines are lines that are not parallel and do not intersect, but also do not lie in the same plane.

To solve this problem, I would start by finding the direction vectors for each line. For line 1, the direction vector is <2, -1, 3>. For line 2, the direction vector is <-1, 3, 2>. For line 3, the direction vector is <2, -1, 3>.

Next, I would use the dot product to determine if any of the direction vectors are parallel. If the dot product is 0, then the lines are perpendicular and therefore not parallel. In this case, the dot product for all three pairs of lines is not 0, so none of the lines are parallel.

To determine if the lines intersect, I would set the parametric equations for each line equal to each other and solve for the parameters. If there is a solution, then the lines intersect at that point. In this case, when solving for t and s, I found that t=3 and s=4, so the lines intersect at the point (7, -4, 9).

Finally, to determine if the lines are skew, I would check if they lie in the same plane. I would take two of the lines and find the vector equation of the plane they lie in. Then, I would check if the third line lies on that plane. If it does not, then the lines are skew. In this case, the third line does not lie on the plane formed by the first two lines, so they are skew.

In summary, the first two lines intersect at the point (7, -4, 9) and all three lines are skew.