Determine whether lines are parallel, intersecting, or skew

In three-dimensional space, lines are classified by comparing direction vectors and solving their parametric equations: distinct lines with proportional direction vectors are parallel, lines sharing a point are intersecting, and nonparallel lines with no common point are skew because they do not lie in one plane. This understanding distinguishes spatial relationships from the two-dimensional case, where skew lines cannot occur, and supports vector methods and analytic geometry without extending to higher-dimensional or projective classifications.