1. The problem statement, all variables and given/known data 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. 2. Relevant equations 3. The attempt at a solution 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.