InnerProductGeometry.inner_eq_zero_iff_angle_eq_pi_div_two
∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] (x y : V),
inner ℝ x y = 0 ↔ InnerProductGeometry.angle x y = Real.pi / 2Two vectors have inner product 0 if and only if the angle between them is π/2.
- Cited by
- 26 results in Mathlib
- Foundations
- Depth 200 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
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
- InnerProductSpacestatement and proof · cited by 3,523
- Real.pistatement · cited by 1,774
- Inner.innerstatement and proof · cited by 1,089
- InnerProductGeometry.anglestatement · cited by 170
- inner_zero_leftproof · cited by 59
- inner_zero_rightproof · cited by 40
Cited by26
Results whose statement or proof uses this declaration.
- EuclideanGeometry.angle_eq_pi_div_two_of_oangle_eq_pi_div_twoproof · cited by 13
- EuclideanGeometry.angle_self_orthogonalProjectionproof · cited by 4
- Orientation.eq_zero_or_oangle_eq_iff_inner_eq_zeroproof · cited by 3
- EuclideanGeometry.angle_eq_arcsin_of_angle_eq_pi_div_twoproof · cited by 3
- EuclideanGeometry.sin_angle_of_angle_eq_pi_div_twoproof · cited by 2
- EuclideanGeometry.angle_lt_pi_div_two_of_angle_eq_pi_div_twoproof · cited by 2
- EuclideanGeometry.dist_div_cos_angle_of_angle_eq_pi_div_twoproof · cited by 2
- EuclideanGeometry.tan_angle_mul_dist_of_angle_eq_pi_div_twoproof · cited by 2
- EuclideanGeometry.tan_angle_of_angle_eq_pi_div_twoproof · cited by 2
- EuclideanGeometry.dist_div_sin_angle_of_angle_eq_pi_div_twoproof · cited by 2
- EuclideanGeometry.dist_div_tan_angle_of_angle_eq_pi_div_twoproof · cited by 2