Meet of lines in n-dimension. I am searching for a general approach to use in a script for determining if two n-dimensional lines represented by one point and their direction vector are skew, parallel, intersecting or identical.
By computing the angle/spread between them I can decide if they are either parallel/identical or skew/intersecting. But to go further I should be able to calculate if there is at least one point which satisfy both of them, which seems easy enough in 3D but I struggle to generalize it into higher dimensions. 
 A: Two meeting lines are in the same plane. And two non parallel lines that lie on a same plane must meet at one and only one point. With this two claims it is enough to see, after checking the given lines are not parallel, if there is a plane containing both of them or not.
Given two non parallel lines $L_1$, defined by point $p_1$ and direction $v_1$, and $L_2$, defined by point $p_2$ and direction $v_2$, calculate $v_3:=p_1-p_2$. Check if the set $\{v_1,v_2,v_3\}$ is linearly independet or not. If it is not, then the lines intersect at one point.
I add a little non-rigorous elaboration of why this works: as $L_1$ and $L_2$ are not parallel the set $\{v_1, v_2\}$ is linearly independent. Then, if the set $\{v_1,v_2,v_3\}$ is linearly dependent that means that there exist two real numbers $\lambda_1$ and $\lambda_2$ such that $\lambda_1 v_1+ \lambda_2 v_2 =v_3=p_1-p_2$. If you stand at $p_2$ and walk $p_1-p_2$ you will reach $p_1$. If you can do this whithout walking out the lines then you will be in the intersection of the lines at some point of your travel. So stand at $p_2$ and walk $\lambda_2 v_2$, you are still in $L_2$ because $L_2$ is defined by direction $v_2$. At this point you know that if now you walk $\lambda_1 v_1$ you will reach $p_1$, but the direction that defines $L_1$ is $v_1$! So now you are in a point that belongs both to $L_1$ and $L_2$.
