Hope anyone can help me with this mathematical problem.
Given 2 sets of 2 lines each in 3D space (where each line is given by a $3D$ point and a unit $3D$ vector): $$ set1=\{ [(px1,py1,pz1), (vx1,vy1,vz1)], [(px2,py2,pz2),(vx2,vy2,vz2)]\} $$ $$ set2=\{ [(qx1,qy1,qz1), (ux1,uy1,uz1)], [(qx2,qy2,qz2),(ux2,uy2,uz2)]\} $$ or in other words: $$ set1=\{line11, line12\} $$ $$ set2=\{line21,line22\} $$ Is it possible that there's no rotation matrix $R$ of size $(3x3)$ such that after rotating $set1$ by $R$, both lines in $set1$ will intersect their corresponding lines in $set2$? Meaning, there's no rotation matrix $R$ such that $line1i$ will intersect $line2i$ ($i \in \{1,2\}$).
I proved that if both sets consisted a single line each, then there's always a rotation matrix that rotates the line in the first set until intersection with the line in the second set (I showed the proof below).
But I'm not sure about 2 sets of 2 lines each. Or at least, I haven't found a contradiction example (two sets of two lines each that the lines in the first set can never intersect both lines in the second set after any rotation).
Can anyone prove/contradict the claim?
Thanks
proof for the one-line claim:
Claim:
given two lines $line1$ and $line2$ (or two sets of one line each) in 3D space. There MUST exist a rotation matrix $R$ such that after rotating $line1$ by $R$, the rotated $line1$ will intersect $line2$.
Proof:
Let $d1$ be the distance between $line1$ and the origin.
Let $d2$ be the distance between $line2$ and the origin.
For every $d3 \geq d1$ there exists a point on $line1$ in which its distance from the origin is $d3$.
For every $d4 \geq d2$ there exists a point on $line2$ in which its distance from the origin is $d4$.
Let $p1$ and $p2$ be points on the lines $line1$ and $line2$ respectively so that the distance between each one of the points from the origin is $d5 \geq max(d1,d2)$.
There is surely a rotation matrix that rotates $p1$ to $p2$ since both points are on the same sphere centered by the origin with radius $d5$. That rotation matrix will end up with intersection between the lines.