Intersecting and skew lines

Objective. Solve parameter systems and classify line relationships.

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Learn: the key idea

Two lines in three dimensions can intersect, be parallel, or be skew, meaning they never meet and are not parallel. To test, set the two vector equations equal, component by component, and solve the resulting system for the parameters; if a consistent solution exists, the lines intersect there, but if the system is inconsistent, the lines are skew (unless the direction vectors are parallel multiples, making them parallel). For r₁ = (1,0,0)+s(1,1,0) and r₂ = (0,1,0)+t(0,1,1), solving the component equations identifies whether a consistent (s,t) pair exists. Always check all three coordinate equations before concluding intersection or skewness.

Worked example

Determine whether the lines r₁ = (1,2,0)+s(1,0,1) and r₂ = (2,2,1)+t(0,1,1) intersect.

1 $Equate components: 1+s = 2, 2 = 2+t, s = 1+t$
2 $From the first equation s = 1; from the second t = 0$
3 $Check the third equation: s should equal 1+t = 1+0 = 1, which matches$
4 $Since all three equations agree with s=1, t=0, the lines intersect$

Answer: The lines intersect, at the point (2,2,1)

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Common mistake. A frequent error is solving only two of the three coordinate equations and assuming the lines intersect, without checking the third equation is also satisfied by the same parameter values. Always verify the solution in all three components.

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