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": $t_1$, $t_3$
Firstly, let $\vec{p_1}, \vec{p_2}, \vec{p_3}, \vec{p_4}$ be the position vector of point $P_1, P_2, P_3, P_4$ respectively. Notice that $P_1$ and $\vec{t_1}$ determines line $L_1$ by $\vec{r_1}(s) = \vec{p_1} + s \vec{t_1}$ where $s \in \mathbb{R}$. Similarly, $P_4$ and $\vec{t_3}$ determines line $L_3$ by $\vec{r_3}(s) = \vec{p_4} + s \vec{t_3}$ where $s \in \mathbb{R}$.
Following the pattern, line $L_2$ can be parameterized as $\vec{r_2}(s) = \vec{p_2} + s \vec{t_2}$ or $\vec{r_2}(s) = \vec{p_3} + s \vec{t_2}$ where $s \in \mathbb{R}$, which one to use is up to you.
So we need at most $2$ more additional parameters, $\{P_2, \vec{t_2}\}$ or $\{P_3, \vec{t_2}\}$. Finally, you can draw some pictures to convince yourself that adding only one of the parameters $P_2, P_3, \vec{t_2}, \alpha_1, \alpha_2$ doesn't determine $L_2$. So $2$ is the minimal number of additional parameters required.
Please tell me if you need help in calculating lengths from these parameters.
EDIT: In the above, we do not consider $L_1, L_2$ and $L_3$ as parameters, but if we do so, then the minimal number of additional parameters required drops down to $1$, because in the argument above, we have already shown that $L_1$ and $L_3$ are in fact determined, thus we only need $L_2$ as an additional parameter to determine all lines.