Distances in space
Distance point–plane with the Hessian normal form, foot of the perpendicular, parallel planes and skew lines.
- France: grade 11
- Germany: grade 12
- Syria: grade 12
Guide
The distance of a point P from a plane E: n₁x₁ + n₂x₂ + n₃x₃ = d is d(P; E) = |n₁p₁ + n₂p₂ + n₃p₃ − d| / |n⃗|. This is the Hessian normal form; |n⃗| is the length of the normal vector.
You find the distance of a point from a line by an auxiliary plane through P perpendicular to g. It meets g in the foot F of the perpendicular, and d = |PF|.
Parallel planes have the same distance everywhere: take a point of one plane and find its distance from the other. For skew lines the common perpendicular lies along n⃗ = u⃗ × v⃗.