I have two points in 3D space, $P_1$ and $P_4$.
From either point, I have a line extended ($L_1$ intersecting $P_1$ and $L_3$ intersecting $P_4$). I want to join these two lines together with another line, $L_2$.
I'm trying to determine the lengths of Lines $L_1$, $L_2$, and $L_3$, but only the direction of lines $L_1$ and $L_3$ are known and given by the unit vectors $t_1$ and $t_3$ respectively. The direction of line $L_2$, or the unit vector $t_2$, is unknown.
I'm not all that great at linear algebra, vector geometry, etc. so correct me if I'm wrong here but I'm guessing I still am missing a variable to define this problem? So let's also say that the angle between $t_1$ and $t_2$ is known as $\alpha_1$ OR that the angle between $t_2$ and $t_3$ is known as $\alpha_2$.
Does all of these parameters make this problem solvable?
EDIT: Thinking about this further it looks like the parameters listed above still leave things under-defined (?) This is a somewhat open ended problem and I can play around with what values are known and unknown. But I'm trying to determine the minimum number required. Values that can be considered known or known: $L_1$, $L_2$, $L_3$, $\alpha_1$, $\alpha_2$
Values that must be considered as "known": $t_1$, $t_3$