Gram-Schmidt Orthonormalization

Orthonormalize a set of vectors

Input vectors

Vector 1
Vector 2
Vector 3
qₖ = (vₖ − Σⱼ ⟨vₖ, qⱼ⟩ qⱼ) / ‖vₖ − Σⱼ ⟨vₖ, qⱼ⟩ qⱼ‖
Orthonormal basis
q₁ = (0.7071, 0.7071, 0)
q₂ = (0.4082, -0.4082, 0.8165)
q₃ = (-0.5774, 0.5774, 0.5774)
Independent vectors
3 of 3
Orthogonality residual
≈ 0
Verified orthonormal
Gram-Schmidt process
  1. v₁ = (1, 1, 0)
    Length ‖u₁‖ = 1.4142
    q₁ = (0.7071, 0.7071, 0)
  2. v₂ = (1, 0, 1)
    − 0.7071·q₁
    Length ‖u₂‖ = 1.2247
    q₂ = (0.4082, -0.4082, 0.8165)
  3. v₃ = (0, 1, 1)
    − 0.7071·q₁ − 0.4082·q₂
    Length ‖u₃‖ = 1.1547
    q₃ = (-0.5774, 0.5774, 0.5774)
Gram-Schmidt OrthonormalizationDiagram of 3 input vectors in 3D and the 3 orthonormal basis vectors produced by Gram-Schmidt, drawn as arrows from the origin.xyzv₁v₂v₃q₁q₂q₃input vᵢorthonormal qᵢ