Theorems · Theorem · geometry

EuclideanGeometry.orthogonalProjection_vsub_mem_direction_orthogonal

∀ {𝕜 : Type u_1} {V : Type u_2} {P : Type u_3} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup V]
  [inst_2 : InnerProductSpace 𝕜 V] [inst_3 : MetricSpace P] [inst_4 : NormedAddTorsor V P] (s : AffineSubspace 𝕜 P)
  [inst_5 : Nonempty ↥s] [inst_6 : s.direction.HasOrthogonalProjection] (p : P),
  ↑((EuclideanGeometry.orthogonalProjection s) p) -ᵥ p ∈ s.directionᗮ

Subtracting p from its orthogonalProjection produces a result in the orthogonal direction.

Defined in

Mathlib.Geometry.Euclidean.Projection

Cited by

4 results in Mathlib

Foundations

Depth 180 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceMetricSpaceNormedAddTorsorNonemptySubmodule.HasOrthogonalProjection