If I have two non-intersecting non-parallel lines in three dimensional space can they define a plane? It's logical to say that if two lines intersect or are parallel that they are on the same plane, but I just can't visualize it the other way around.
Any help is appreciated.
On
Two lines $r$ and $s$ in ${\Bbb R}^3$ that do not meet in a point and are not parallel are never contained in a plane.
But you can play the following game: write parametric equations $r(t)$ and $s(t)$ for $r$ and $s$ respectively and for any value of $t\in\Bbb R$ let $\ell_t$ the line joining the point $r(t)$ and the point $s(t)$ (well defined, since $r$ and $s$). As $t$ varies, the line $\ell_t$ describe a surface $S$ which may have some interesting properties (when $r$ and $s$ are parallel not coincident, $S$ is the plane where the two lines sit in).
It is not necessary for those $2$ lines to define a plane: consider for example the $x$-axis and the line defined by shifting the $y$ axis by $1$ in the $z$-direction