Gram-Schmidt Orthogonalization in 2D

Input Vectors (drag the arrowheads)

(2.20, 0.60)

(1.00, 1.80)

Animation speed 1.0×

Orthonormal basis

e1 · e20.000

|e1|1.000

|e2|1.000

How it works

The Gram-Schmidt process turns a set of linearly independent vectors {v1, v2 (, v3)} into an orthonormal basis {e1, e2 (, e3)} — vectors that are mutually perpendicular and exactly unit length — by repeatedly subtracting off the part of each new vector that already lies along the ones already processed:

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|

Drag the tips of v1 and v2 anywhere on the plane; e1 and e2 update live, and "Animate steps" plays the actual computed projection, subtraction and normalization for the current vectors, one operation at a time.

Drag v1 / v2 (/ v3) — every stat and arrow recomputes from the real dragged coordinates, no canned animation.
Show projections — draws proj_u1(v2) (and proj_u1(v3), proj_u2(v3)) as dashed arrows, the exact quantities the formula subtracts, plus the residual u2 (u3).
e1 · e2, e1 · e3, e2 · e3 — the orthogonality check: a perfect basis gives exactly 0; floating-point arithmetic gives something on the order of 1e-16.
|e1|, |e2|, |e3| — the norm check: every output vector should measure exactly 1.
Add third vector — because the plane is only 2-dimensional, any third vector is automatically a linear combination of the first two. Watch u3 collapse toward the zero vector as you drag v3 around — that collapse is the algorithm correctly detecting linear dependence, not a bug.

This is the algorithm behind QR decomposition, used throughout numerical linear algebra (least-squares fitting, eigenvalue solvers, computer-graphics camera bases) to replace a numerically messy set of vectors with a clean orthonormal frame.

Drag an arrowhead to move that vector

v1 v2 projection (dashed) e1, e2 (, e3) orthonormal