Gram-Schmidt Calculator
Perform Gram-Schmidt orthogonalization to create orthogonal and orthonormal vector sets.
Input Vectors
v1
v2
v3
Orthonormal Basis
e1 = (0.7071, 0.7071, 0.0000)
e2 = (0.4082, -0.4082, 0.8165)
e3 = (-0.5774, 0.5774, 0.5774)
Orthogonal Vectors (unnormalized)
u1 = (1.0000, 1.0000, 0.0000)
u2 = (0.5000, -0.5000, 1.0000)
u3 = (-0.6667, 0.6667, 0.6667)
Orthogonality Check
e1 · e2 = 0.0000000000
e1 · e3 = 0.0000000000
e2 · e3 = 0.0000000000
Normalization Check
|e1| = 1.0000000000
|e2| = 1.0000000000
|e3| = 1.0000000000
Gram-Schmidt Process
Step 1: Orthogonalize
u₁ = v₁
uₖ = vₖ - Σ proj(vₖ onto uⱼ)
Step 2: Normalize
eₖ = uₖ / |uₖ|