Parallel, intersecting and skew lines
Two lines in 3D may be parallel, intersecting or skew. Check direction first, then test whether one pair of parameter values gives the same point in all three components.
Use a two-stage classification
Classify lines in two stages
same / parallel
one = scalar × other
then test an anchor
intersect
one parameter pair
all 3 components agree
skew
directions not parallel
no common point
Use different parameters for the two lines.
- Compare the direction vectors. If they are proportional, meaning one is a scalar multiple of the other, the lines are parallel or the same line. Substitute one anchor point into the other line. If one parameter value gives all three coordinates, the lines are the same; otherwise they are distinct parallel lines.
- If the directions are not proportional, equate the position vectors using different parameter letters.
- Solve two component equations, then check the third. A failed third equation means the lines are skew: not parallel, but with no common point.
For example, lies on because gives all three coordinates. A line through that point with direction is therefore the same line, not a separate parallel line.
Common mistake
Solve two components and verify the third
Worked example
Current-paper intersection
Equating components gives:
These give and . The third component checks:
(9709/32/M/J/24 Q8(b))
Keep the two parameters separate
Find the intersection of and , or state why no intersection exists.
Show worked answer
The second component gives . Substitution in the first component gives . The first line then has , but the second has .