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

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

• if $(x_1, y_1, z_1)$ is on $l_2$, 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:

We pick two and solve for $\lambda$ and $\mu$.

• If they hold for the third equation, then there is an intersection! Substitute either $\lambda$ or $\mu$ 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 $t$ as a parameter. Here a single $t$ must simultaneously solve

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