A line in 3D is determined by a point and a direction, or by two points. The three standard representations are:

Symmetric (Cartesian) form: $\frac{x-x1}{a}$ = $\frac{y-y1}{b}$ = $\frac{z-z1}{c}$, where (x1,y1,z1) is a point on the line and (a,b,c) are direction ratios. If any DR is zero (say c=0), the convention is $\frac{x-x1}{a}$ = $\frac{y-y1}{b}$, z = z1.

Vector form: r = a + t*b, where a is the position vector of a point and b is the direction vector. The parameter t ranges over all reals.

Two-point form: $\frac{x-x1}{(x2-x1)}$ = $\frac{y-y1}{(y2-y1)}$ = $\frac{z-z1}{(z2-z1)}$, passing through (x1,y1,z1) and (x2,y2,z2).

A line can also be defined as the intersection of two planes: a1x+b1y+c1z=d1 and a2x+b2y+c2z=d2. The direction of such a line is n1 x n2 (cross product of the normals).

Key results: Two lines may be intersecting, parallel, or skew (non-parallel, non-intersecting). Two lines are coplanar iff the scalar triple product [a2-a1, b1, b2] = 0.

The shortest distance between skew lines r=a1+tb1 and r=a2+sb2 is SD = |(a2-a1).(b1 x b2)| / |b1 x b2|. For parallel lines with direction b, SD = |(a2-a1) x b| / |b|.

The perpendicular distance from a point P to a line through A with direction b is d = |AP x b| / |b|. The foot of the perpendicular is found by setting AP.b = 0 for the parametric point on the line.