Intersection of 3D lines (HL)
There are 4 cases:
- Identical lines
- Parallel lines
- Intersecting lines
- Skew lines (ie lines on parallel planes)
Let’s say we are asked to find the intersection(s), if any, of the lines
If is a scalar multiple of , and
- if is on , 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 and .
- If they hold for the third equation, then there is an intersection! Substitute either or 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 as a parameter.
Two linear, constant-velocity trajectories meet if and only if there is a that solves all three equations. See also distances between lines and between trajectories