Transform vectors into stable orthogonal directions. Inspect norms, projections, and QR structure instantly and clearly. Solve basis problems with transparent stepwise calculations online today.
| Vector | Input Components | Orthogonal Output | Status |
|---|---|---|---|
| v1 | [1, 1, 0] | [1, 1, 0] | Independent |
| v2 | [1, 0, 1] | [0.5, -0.5, 1] | Independent |
| v3 | [0, 1, 1] | [-0.666667, 0.666667, 0.666667] | Independent |
The calculator applies the classical Gram Schmidt process to each input vector.
u1 = v1
uk = vk - Σ[(vk · ui) / (ui · ui)] ui
ek = uk / ||uk||
Q = [e1 e2 ... er]
R = QᵀA
Here, A is the original vector matrix. Q stores orthonormal directions. R stores the projection weights. Near zero remainders reveal linear dependence.
Gram Schmidt decomposition turns overlapping vectors into orthogonal directions. That is valuable in physics. Many physical systems start with mixed coordinates. Basis states can overlap. Measurement axes can be coupled. This calculator separates those directions with a clean and structured process.
It computes orthogonal vectors, orthonormal vectors, compact Q and R matrices, norms, and dependence checks. These outputs help with quantum state preparation, mode analysis, coordinate transforms, field projections, and signal space interpretation. The tool also shows each subtraction step. That improves trust in the result.
In quantum mechanics, orthonormal basis vectors simplify inner products and probability amplitudes. In classical mechanics, they help describe constrained motion and rotated frames. In electromagnetism, they support projections of fields onto independent directions. In numerical physics, they improve matrix factorization workflows and stable simulations.
This Gram Schmidt decomposition calculator is useful when you need to verify whether vectors are independent. It also helps when you must build a compact QR decomposition from measured or theoretical data. The rank output quickly shows how much real information exists in the input set.
The orthogonal basis shows the cleaned directions before normalization. The orthonormal basis shows unit vectors with the same span. The Q matrix stores those unit directions. The R matrix stores the projection magnitudes. If a remainder becomes zero within tolerance, the calculator marks the vector as dependent.
That matters for physics models. Dependent vectors do not add new dimensions. They only repeat existing information. By removing redundancy, you get a clearer basis for analysis, simulation, and interpretation. This makes the calculator practical for students, researchers, and engineers who work with vector spaces every day.
It returns orthogonal vectors, optional orthonormal vectors, compact Q and R matrices, vector norms, rank, dependence checks, and stepwise decomposition details.
It builds independent basis directions for state spaces, field components, mode shapes, and coordinate systems. That makes projections and matrix analysis easier.
If a remainder becomes nearly zero, the calculator marks that vector as dependent. It does not create a new basis direction from redundant input.
Tolerance controls how the calculator treats very small remainders. It helps with floating point rounding and practical dependence decisions.
Orthogonal vectors are mutually perpendicular. Orthonormal vectors are also perpendicular, but each vector has unit length.
Yes. It forms a compact Q matrix from orthonormal basis vectors and computes a compact R matrix using the original input vectors.
Yes. Extra vectors are allowed. Any vector that does not add a new independent direction will be flagged as dependent.
Yes. After calculation, you can download the generated result tables as CSV or PDF for documentation or sharing.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.