Intersection of 3D lines (HL)

There are 4 cases:

  1. Identical lines
  2. Parallel lines
  3. Intersecting lines
  4. Skew lines (ie lines on parallel planes)

Let’s say we are asked to find the intersection(s), if any, of

r1=(l,m,n)λ+(x1,y1,z1)\bm{r_1} = (l, m, n)\lambda + (x_1, y_1, z_1)
r2=(p,q,s)μ+(x2,y2,z2)\bm{r_2} = (p, q, s)\mu + (x_2, y_2, z_2)

If (l,m,n)(l, m, n) is a scalar multiple of (p,q,s)(p, q, s), and

  • if (x1,y1,z1)(x_1, y_1, z_1) is on l2l_2, then the two lines are identical
  • otherwise, the two lines are parallel

Otherwise it’s either intersecting or skew.

At the point(s) of intersection:

x:    lλ+x1=pμ+x2x:\;\; l\lambda + x_1 = p\mu + x_2
y:    mλ+y1=qμ+y2y:\;\; m\lambda + y_1 = q\mu + y_2
z:    nλ+z1=sμ+z2z:\;\; n\lambda + z_1 = s\mu + z_2

We pick two and solve for λ\lambda and μ\mu.

  • If they hold for the third equation, then there is an intersection! Substitute either λ\lambda or μ\mu into the original equations to find the point of intersection
  • otherwise, they are skew lines

There are other methods to find the point of intersection, but they involve cross products and/or magnitudes, hence larger numbers, so they are arguably harder. 

Sometimes, the two lines are linear trajectories. These typically involve time tt as a parameter. Here a single tt must simultaneously solve

x:    lt+x1=pt+x2x:\;\; lt + x_1 = pt + x_2
y:    mt+y1=qt+y2y:\;\; mt + y_1 = qt + y_2
z:    nt+z1=st+z2z:\;\; nt + z_1 = st + z_2

The differ from lines mostly when we consider distances between lines and between trajectories