Find the shortest distance between the lines vector r = (4i - j) + λ(i + 2j - 3k) and vector r = (i - j + 2k) + μ(2i + 4j - 5k).

Question

Find the shortest distance between the lines vector r = (4i - j) + λ(i + 2j - 3k) and vector r = (i - j + 2k) + λ(2i + 4j - 5k).

Answer 1

Comparing the given equation with the standard equatiary.

vector r = vector(a1 + λb1) and vector r = vector(a2 + λb2)

vector a1 = 4i - j, vector b1 = i + 2j - 3k

vector a2 = i - j + 2k, vector b2 = 2i + 4j - 5k

∴ vector(a2 - a1) = i - j + 2k - 4i + j = -3i + 2k

∴ vector(b1 x b2) =

= i(-10 + 12) - j(-5 + 6) + k(4 - 4) = 2i - j

∴ |vector(b1 x b2)| = √(4 + 1) = √5

∴ *.D.