 Joint of a line collection

For distinct real affine lines in ${\mathbb{R}}^n$, $n\ge 2$, a joint of a line collection is a point lying on $n$ lines with linearly independent direction vectors. In three dimensions the directions must not all lie in one plane. Extra incident lines are allowed; existence of one spanning choice is enough. This counts joint of a line collection points, rather than tuples of lines or incidence multiplicities.