Gram-Schmidt Orthogonalization — Building an Orthonormal Basis

Gram-Schmidt Process

v₃ · x 0.60

v₃ · y 0.40

v₃ · z 1.80

Show orthonormal frame e₁,e₂,e₃

u₁ · u₂

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u₁ · u₃

–

u₂ · u₃

–

det [u₁ u₂ u₃]

–

How it works

Three ordinary vectors v₁, v₂, v₃ point in independent but non-perpendicular directions (pale arrows). The Gram-Schmidt process turns them into a mutually perpendicular set u₁, u₂, u₃ (bold arrows) one step at a time: keep v₁ as u₁, then strip out of v₂ whatever component points along u₁, then strip out of v₃ whatever points along both u₁ and u₂. What is left after each subtraction is, by construction, perpendicular to everything already built.

proj_u(v) = (v·u / u·u) · u

u1 = v1
u2 = v2 − proj_u1(v2)
u3 = v3 − proj_u1(v3) − proj_u2(v3)

e_i = u_i / |u_i|   (orthonormal basis)

v₃ sliders — move the third input vector; u₁ and u₂ don't change (they never depended on v₃), but u₃ recomputes live and the dashed projection lines re-aim toward it.
Projections toggle — shows the dashed proj_u1(v2), proj_u1(v3) and proj_u2(v3) segments being subtracted at each step.
Orthonormal frame — divides each u_i by its own length, shrinking the bold arrows to unit vectors e₁,e₂,e₃ — the actual output of QR decomposition's Q matrix.
Dot-product readouts — u_i·u_j should read 0.000 for every pair regardless of the sliders; that's the process working. The determinant of [u₁ u₂ u₃] equals the determinant of [v₁ v₂ v₃] — Gram-Schmidt preserves volume, it only re-squares the frame.

This is exactly the linear-algebra step underneath QR decomposition, used to numerically stabilize least-squares regression, and underneath re-orthonormalizing a rotation/camera basis that has drifted from floating-point error.

v₁ / u₁ (unchanged) v₂ → u₂ v₃ → u₃ projection being subtracted

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