InnerProductGeometry.angle_eq_pi_iff
∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] {x y : V},
InnerProductGeometry.angle x y = Real.pi ↔ x ≠ 0 ∧ ∃ r < 0, y = r • xThe angle between two vectors is π if and only if they are nonzero and one is a negative multiple of the other.
- Cited by
- 8 results in Mathlib
- Foundations
- Depth 197 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites13
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Realstatement and proof · cited by 25,697
- NormedAddCommGroupstatement and proof · cited by 15,752
- Norm.normproof · cited by 5,413
- InnerProductSpacestatement and proof · cited by 3,523
- Real.pistatement and proof · cited by 1,774
- Inner.innerproof · cited by 1,089
- InnerProductGeometry.anglestatement · cited by 170
- Real.arccosproof · cited by 90
- abs_leproof · cited by 64
- LE.le.ge_iff_eq'proof · cited by 50
- abs_real_inner_div_norm_mul_norm_le_oneproof · cited by 4
- real_inner_div_norm_mul_norm_eq_neg_one_iffproof · cited by 2
Cited by8
Results whose statement or proof uses this declaration.
- Sbtw.angle₁₂₃_eq_piproof · cited by 6
- EuclideanGeometry.angle_eq_pi_iff_sbtwproof · cited by 6
- InnerProductGeometry.angle_eq_angle_add_add_angle_addproof · cited by 2
- EuclideanGeometry.angle_eq_angle_of_angle_eq_piproof · cited by 2
- InnerProductGeometry.angle_add_angle_eq_pi_of_angle_eq_piproof · cited by 1
- EuclideanGeometry.angle_eq_zero_of_angle_eq_pi_leftproof · cited by 1
- Orientation.angle_eq_iff_oangle_eq_or_sameRayproof · cited by 1