Lesson 6 of 9 · 6 min
Distance between parallel lines
NCERT §11.5.2
A fourth laser from the side wall runs parallel to the first, through A(3, 7, 5). How far apart are the two parallel beams?
The lesson in notes
In short
Parallel lines lie in one plane, and the distance between them is the length of a perpendicular from a point of one to the other.
For r = a₁ + λb and r = a₂ + μb (the same direction b) the distance is d = |b × (a₂ − a₁)| / |b|: the joining vector's component perpendicular to b.
Why: |b × (a₂ − a₁)| = |b| |a₂ − a₁| sin θ, and |a₂ − a₁| sin θ is exactly the perpendicular gap.
The skew-line formula fails here because b₁ × b₂ = 0; use this one instead.
Worked: r = î + 2ĵ − 4k̂ + λ(2î + 3ĵ + 6k̂) and r = 3î + 3ĵ − 5k̂ + μ(2î + 3ĵ + 6k̂). Then a₂ − a₁ = 2î + ĵ − k̂, b × (a₂ − a₁) = −9î + 14ĵ − 4k̂ of length √293, and |b| = 7, so d = √293/7.