Find two lines in R3 in parametric form which satisfy the following conditions. Also, find the points on the lines which achieve the closest distance.

Conditions:
1. They are not parallel to any of the coordinate planes
2. They do not intersect and are not parallel
3. They are a distance of 2 apart at the points which have parameter values t = 0 and u = 0
4. The lines are NOT closest when the parameter values are t = 0 and u = 0.

My attempt:
I know how to find the shortest distance between two skew lines, but how do I find where that distance occurs in terms of points on the lines if all I have are parameters? I'm thinking that I need to use projections from the lines onto a plane then find the intersection of the projections, but when I try to do this I either get an unsolvable equation or infinite number of answers, since I'm dealing with parameters rather than exact coordinates.

This problem would be easier if the closest distance was 2 and occurred when t = 0 and u = 0, since I could start with two lines distance 2 apart parallel to the xz plane, then rotate them around the z axis 45 degrees using a rotation matrix. If the closest distance was 2 and occurred when t = 0 and u = 0, my lines could be (-t*sqrt(2)/2, -t*sqrt(2)/2, t) and (u*sqrt(2)/2 - sqrt(2), u*sqrt(2)/2 + sqrt(2), u) but unfortunately that is not the case here.

In terms of the distances, for 2 lines
L1 = (a,b,c) + t[v1,v2,v3]
L2 = (d,e,f) + u[w1,w2,w3]

distance between them must be 2:

2 = sqrt((a-d)^2 + (b-e)2 + (c-f)^2)
so 4 = (a-d)^2 + (b-e)2 + (c-f)^2

2 sets of lines I tried:

(0,0,2) + t[3,1,-1]
(0,0,0) + u[1,2,3]

for these the shortest distance seems to be 2/sqrt(6), which is indeed less than 2, but where does this distance occur? and

(0,0,3) + t[1,2,-1]
(0,0,1) + u[1,-1,1]

while for these its 6/sqrt(14), but I still don't see how I can find where this distance occurs.

Question

Find two lines in R3 in parametric form which satisfy the following conditions. Also, find the points on the lines which achieve the closest distance.

Conditions:
1. They are not parallel to any of the coordinate planes
2. They do not intersect and are not parallel
3. They are a distance of 2 apart at the points which have parameter values t = 0 and u = 0
4. The lines are NOT closest when the parameter values are t = 0 and u = 0.

My attempt: 
I know how to find the shortest distance between two skew lines, but how do I find where that distance occurs in terms of points on the lines if all I have are parameters? I'm thinking that I need to use projections from the lines onto a plane then find the intersection of the projections, but when I try to do this I either get an unsolvable equation or infinite number of answers, since I'm dealing with parameters rather than exact coordinates.

This problem would be easier if the closest distance was 2 and occurred when t = 0 and u = 0, since I could start with two lines distance 2 apart parallel to the xz plane, then rotate them around the z axis 45 degrees using a rotation matrix. If the closest distance was 2 and occurred when t = 0 and u = 0, my lines could be (-t*sqrt(2)/2, -t*sqrt(2)/2, t) and (u*sqrt(2)/2 - sqrt(2), u*sqrt(2)/2 + sqrt(2), u) but unfortunately that is not the case here.

In terms of the distances, for 2 lines 
L1 = (a,b,c) + t[v1,v2,v3]
L2 = (d,e,f) + u[w1,w2,w3]

distance between them must be 2:

2 = sqrt((a-d)^2 + (b-e)2 + (c-f)^2)
so 4 = (a-d)^2 + (b-e)2 + (c-f)^2

2 sets of lines I tried:

(0,0,2) + t[3,1,-1]
(0,0,0) + u[1,2,3]

for these the shortest distance seems to be 2/sqrt(6), which is indeed less than 2, but where does this distance occur? and

(0,0,3) + t[1,2,-1]
(0,0,1) + u[1,-1,1]

while for these its 6/sqrt(14), but I still don't see how I can find where this distance occurs.

ash2326 · Answer

wow, good attempt:)
Sorry, I don't know this
@satellite73 sir @lgbasallote @Callisto Please help here

callisto · Answer

I'm sorry that I haven't learnt this topic yet :(

ash2326 · Answer

@UnkleRhaukus help here

apoorvk · Answer

W-w-wait.#2 at the very top says "They do not intersect and are not parallel" - how's that earthly possible? :o
@Eyes5120 could you please check if there's a typo till I read h rest of it? Thanks.

kinggeorge · Answer

@apoorvk it's in 3 dimensions, so the lines may be skew. As for actually doing this, problem, my recommendation would be to choose one line to start at the origin at t,u=0, and the other line to start at (2,0,0) at t,u=0. From that, construct some lines that satisfy the conditions, and then find the equations.

experimentx · Answer

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