The lines `l_1` and `l_2` have equations `r = 2i+k+\lambda(i-j+2k)` and `r = 2j+6k+\mu(i+2j-2k)` respectively. The point P on `l_1` and the point Q on `l_2` are such that PQ is perpendicular to both `l_1` and `l_2`.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2022). The mark or mark type is shown in brackets.

1. [B1] Finds a vector joining any point of `l_1` to any point of `l_2`. (e.g. `(0,2,6)-(2,0,1) = (-2,2,5)`.)
2. [M1 A1] Finds common perpendicular vector. (e.g. `d₁ × d₂ = (1,-1,2) × (1,2,-2) = (-2,4,3)`.)
3. [M1 A1] Uses formula for shortest distance. (`| (a₂-a₁) · (d₁ × d₂) | / |d₁ × d₂| = 27/√29 ≈ 5.01`.)
4. [B1 FT] `r = (2,0,1) + s(1,-1,2) + t(-2,4,3)` (FT their common perpendicular from part (a).)
5. [M1 A1 FT] Finds normal to the plane `Pi₂`. (`n₂ = d₂ × PQ = (1,2,-2) × (-2,4,3) = (14,1,8)` (FT their common perpendicular from part (a).)
6. [M1 A1] Substitutes point. (`14(0) + (2) + 8(6) = 50 \implies 14x+y+8z = 50`.)
7. [M1 A1 FT] Finds normal to the plane `Pi₁`. (`n₁ = d₁ × PQ = (1,-1,2) × (-2,4,3) = (-11,-7,2)` (FT their part (b)(i).)
8. [M1 A1] Uses dot product of normal vectors. (`cosθ = (|n₁ · n₂|)/(|n₁||n₂|) = (|(-11)(14)+(-7)(1)+(2)(8)|)/√174√261 = 145/√174√261`.)
9. [A1] `47.1^{\circ}` or `0.822` radians.