Introduction: Lines in Three-Dimensional Space
In two-dimensional geometry, two distinct lines either intersect or are parallel , those are your only options. But in three-dimensional space, something more interesting is possible: lines can miss each other without being parallel at all.
This subtopic (AHL 3.15) formalises the four possible relationships between two lines in 3D, and develops the techniques to classify and work with them. Mastering this is essential not just for geometry questions, but also connects directly to solving systems of linear equations (AHL 1.16).
All lines in this subtopic are expressed in vector form: where is a position vector of a point on the line, is the direction vector, and is the parameter.
The Four Classifications of Lines in 3D
Two lines in three-dimensional space must fall into exactly one of the following four categories:
- Coincident , the lines are identical (occupy exactly the same set of points)
- Parallel , the lines have proportional direction vectors and never meet
- Intersecting , the lines cross at exactly one point
- Skew , the lines are not parallel and do not intersect
Think of two roads on a map (2D) , they either cross or run parallel. Now imagine roads in a city with flyovers: one road can pass over another without ever touching it, and they're not even going in the same direction. That's a skew relationship , it only exists in 3D.
Skew lines are unique to 3D (and higher-dimensional) geometry. In 2D, non-parallel lines always intersect.
