Gram-Schmidt Orthonormalization

Orthonormalize a set of vectors

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

v₁ = (1, 1, 0)
Length ‖u₁‖ = 1.4142
q₁ = (0.7071, 0.7071, 0)
v₂ = (1, 0, 1)
− 0.7071·q₁
Length ‖u₂‖ = 1.2247
q₂ = (0.4082, -0.4082, 0.8165)
v₃ = (0, 1, 1)
− 0.7071·q₁ − 0.4082·q₂
Length ‖u₃‖ = 1.1547
q₃ = (-0.5774, 0.5774, 0.5774)

About this calculator

The Gram-Schmidt Orthonormalization Calculator turns up to four 2D or 3D vectors into an orthonormal basis using the modified Gram-Schmidt process in 30-digit decimal arithmetic. Every step is shown: the projection coefficients removed from each vector, the orthogonal residual, its norm, and the final unit vector. Linearly dependent inputs are detected and dropped (reducing the rank), already-orthonormal sets are recognized, and a mutual-orthogonality residual verifies the result.

How to use the Gram-Schmidt Orthonormalization calculator

Enter the values for your problem into the input fields.
Read the result — it updates instantly as you type.
Check the formula and the visual explanation to follow how the answer was found.
Copy the page URL to share the exact calculation.

Common examples

(3, 1), (2, 2) → q₁ ≈ (0.948683, 0.316228), q₂ ≈ (−0.316228, 0.948683)
(1,1,0), (1,0,1), (0,1,1) → full 3D orthonormal basis, q₁ ≈ (0.707107, 0.707107, 0)
(1, 2) and (2, 4) → second vector is dependent and dropped, rank 1
(1, 0), (0, 1) → recognized as already orthonormal, returned unchanged

Frequently asked questions

How does the Gram-Schmidt Orthonormalization calculator work?

Enter your values and the calculator applies the exact mathematical method for this problem, showing the result together with the formula it used. Everything is computed with high-precision arithmetic, so the answer you see is not limited by ordinary floating-point rounding.

When would I use the Gram-Schmidt Orthonormalization calculator?

It is useful for homework and exam preparation, for checking work done by hand, and whenever a step in a larger problem needs this calculation done quickly and reliably.

How accurate are the results?

Calculations run at 30 significant digits of precision internally. Displayed values are rounded for readability, but the underlying result is far more precise than a typical hand or pocket-calculator computation.

Can I share a specific calculation with someone else?

Yes. Every value you enter updates the URL, so copying the address bar and sharing the link reopens the exact same calculation for whoever clicks it.