With respect to a fixed origin O, the lines L₁ and L₂ are given by the equations: L₁: r = ⟨1, 2, -1⟩ + λ⟨-2, 1, 3⟩ L₂: r = ⟨-1, 1, 4⟩ + μ⟨0, 1, -1⟩ (b) Find the shortest distance between L₁ and L₂.

Mark scheme

Marking points from the official mark scheme. The mark or mark type is shown in brackets.

1. M1 for finding the vector connecting two general points on L₁ and L₂: d = ⟨-1 - (1 - 2λ), 1 - (2 + λ), 4 - (-1 + 3λ)⟩ = ⟨-2 + 2λ, -1 - λ + μ, 5 - 3λ - μ⟩.
2. M1 for using the condition that d is perpendicular to both direction vectors, ie. d ⋅ ⟨-2, 1, 3⟩ = 0 and d ⋅ ⟨0, 1, -1⟩ = 0.
3. A1 for forming the two correct equations: (-2+2λ)(-2) + (-1-λ+μ)(1) + (5-3λ-μ)(3) = 0 => 16 - 14λ - 2μ = 0 AND (-1-λ+μ)(1) + (5-3λ-μ)(-1) = 0 => -6 + 2λ + 2μ = 0.
4. M1 for solving these simultaneous equations for λ and μ.
5. A1 for λ = 1 and μ = 2.
6. A1 for calculating the shortest distance using the values of λ and μ, i.e., finding the magnitude of d or finding the distance between the two points. Distance = |⟨0, 0, 0⟩ - ⟨-1, -1, 1⟩| = |⟨1, 1, -1⟩| = √3.