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2D Vector Linear Combination Calculator — result sheet
Solve for the two coefficients that express a target 2D vector as a linear combination of two supplied vectors.
Inputs used
Results
Visual chart
Breakdown
Calculation steps
Returned data table
Formula and methodology
Formula: Solve [u v] [alpha beta]ᵀ=[target x target y]ᵀ; determinant=u_x v_y−v_x u_y; alpha=(t_x v_y−v_x t_y)/det; beta=(u_x t_y−t_x u_y)/det.
The two supplied vectors form a coordinate basis only when they are not parallel. This calculator solves the resulting 2×2 system for alpha and beta, then reconstructs the target components so the visitor can check that alpha times the first vector plus beta times the second vector returns the entered target.
This result follows the calculator's declared inputs, precision, validation boundaries, and model limits.
Input contract
- First vector x — minimum -1000000000; maximum 1000000000
- First vector y — minimum -1000000000; maximum 1000000000
- Second vector x — minimum -1000000000; maximum 1000000000
- Second vector y — minimum -1000000000; maximum 1000000000
- Target vector x — minimum -1000000000; maximum 1000000000
- Target vector y — minimum -1000000000; maximum 1000000000
Worked example
| Input | Value |
|---|---|
| First vector x | 1 |
| First vector y | 0 |
| Second vector x | 0 |
| Second vector y | 1 |
| Target vector x | 3 |
| Target vector y | 4 |
alpha = 3; beta = 4; determinant = 1; reconstruction = (3, 4).
Assumptions and limits
- All six components are finite real numbers on the same 2D coordinate system.
- The determinant must be nonzero, so the two basis vectors cannot be parallel or zero in a way that makes the system singular.
- The coefficients are exact algebraic solutions displayed as finite decimals; no rounding is used before reconstruction.
- The page solves a 2D two-vector system and does not infer a best fit for an overdetermined data set or a higher-dimensional basis.
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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