Topic 3 · IB Maths AA
Vector equations of lines in 3D
Build two lines r = a + λb from integer vectors, rotate the 3D view, and see whether they intersect, are parallel or are skew, together with the acute angle between them.
Key equations
r = a + λb
cos θ = |b₁·b₂| / (|b₁||b₂|)
Skew distance = |(a₂ − a₁)·(b₁×b₂)| / |b₁×b₂|
Try this
- Choose Intersecting, then set b₂ = 2b₁. The lines become parallel and θ = 0°; now change a₂ until the lines are coincident.
- Choose Skew and change one component of a₂ at a time until (a₂ − a₁)·(b₁×b₂) reaches 0. What happens to the classification?
- Find your own b₁ and b₂ with b₁·b₂ = 0 and check that the angle reads 90.0°.
Exam tip: after solving two component equations for λ and μ, always substitute into the third — if it fails and the direction vectors are not parallel, the lines are skew.