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Two-Dimensional Gram-Schmidt Calculator — result sheet
Orthogonalize two independent two-dimensional vectors and return an orthonormal basis.
Inputs used
Results
Visual chart
Breakdown
Calculation steps
Returned data table
Formula and methodology
Formula: e1=u/||u||; v⊥=v−(v·e1)e1; e2=v⊥/||v⊥||.
The projection and residual norm are shown so the second orthonormal vector can be checked directly.
This result follows the calculator's declared inputs, precision, validation boundaries, and model limits.
Input contract
- Vector u x — minimum -1000000; maximum 1000000
- Vector u y — minimum -1000000; maximum 1000000
- Vector v x — minimum -1000000; maximum 1000000
- Vector v y — minimum -1000000; maximum 1000000
Worked example
| Input | Value |
|---|---|
| Vector u x | 1 |
| Vector u y | 1 |
| Vector v x | 1 |
| Vector v y | -1 |
e1≈(0.7071,0.7071); e2≈(0.7071,-0.7071).
Assumptions and limits
- The first vector is nonzero and the two vectors are linearly independent.
- Euclidean dot product and norm are used.
- The sign of an orthonormal basis vector may differ from another valid orientation.
Calculator note
Source and methodology
Use the official WorldCalculate methodology policy for the source, formula, precision, and boundary standards behind this calculator.
Planning estimate, not financial, medical, legal, or professional advice. © WorldCalculate — reuse with attribution. Built and curated by Hassan ALRowaie.
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