1.5.2 Directed Lines
Directed lines in are not substantially different than the ones we studied in . Just as in we can parameterize a line via a point on the line and a vector describing the way the line changes from one point to the next.
Parameterize the line containing the two points and .
One subtlety about lines in space:
In the planes any two distinct lines are either intersecting or parallel. In space we pick up a third option - skew.
- Intersecting lines will share one point. It's important to note that if and are parameterizations of two intersecting lines that there is not necessarily a single value for which . When that happens we say the parameterizations collide. In general you would expect the point of intersection to result from different values of the parameter for each of the two lines.
- Parallel lines will have the same slope vector.
- Skew lines are lines that do not have the same slope vector but also never intersect. You need at least three dimensions to have skew lines (so no pair of lines in the plane can ever be skew).
