Gram-Schmidt Orthogonalization Calculator

e₁ (first orthonormal vector)

(0.2673, 0.8018, -0.5345)

First basis vector normalized to unit length.

e₂ (second orthonormal vector)(0.4438, 0.39, 0.8068)

e₃ (third orthonormal vector)(-0.8554, 0.4528, 0.2516)

u₁ (first orthogonal vector)(1, 3, -2)

u₂ (second orthogonal vector)(2.3571, 2.0714, 4.2857)

u₃ (third orthogonal vector)(-2.9266, 1.5494, 0.8608)

||u₁||3.7417

||u₂||5.3117

||u₃||3.4215

||u₁||3.7417

||u₂||5.3117

||u₃||3.4215

Orthonormal basis computed for 3 vectors in 3D space.

Each output vector e_k has length exactly 1 and is perpendicular to every other output vector.
The set of e_k vectors spans the same subspace as your original input vectors.
You can verify orthonormality by checking that e_i · e_j = 0 for i ≠ j, and ||e_k|| = 1 for all k.
With 3 vectors in 3D, the result spans all of R3 if the vectors are independent.

Next stepThese orthonormal vectors can be used as columns of an orthogonal matrix Q in QR decomposition, or as a stable numerical basis for projections.

Gram-Schmidt Orthogonalization Calculator

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