Linear Combination of Vectors Calculator
This calculator computes the linear combination of several vectors weighted by scalar coefficients, returning the resulting vector and its magnitude.
Resulting vector: [1, 2]
- 1
Result vector r = c₁·v₁ + … + cₖ·vₖ
[1, 2]Each component j is Σᵢ cᵢ·vᵢ[j]. - 2
‖r‖² (sum of squared components)
1² + 2² = 5 - 3
Magnitude ‖r‖ = √(‖r‖²)
√(5) = 2.2361
How does this calculator work?
A linear combination scales each vector by its coefficient and adds them: r = c₁v₁ + c₂v₂ + ... + cₖvₖ, computed component by component. Enter vectors as rows and a matching list of scalars; the calculator returns the resulting vector, its dimension, and its magnitude, and plots 2D inputs.
Formula
How this is calculated
Enter your vectors one per row in the first box. Every vector must have the same number of components (the dimension d); for example two-dimensional vectors are written as "x, y". In the second box enter one scalar coefficient per vector, in the same order, separated by commas or spaces.
The calculator multiplies each vector by its matching scalar and adds the results component by component: for each position j the result is r[j] = c₁·v₁[j] + c₂·v₂[j] + ... + cₖ·vₖ[j]. The output is a single vector of the same dimension d, plus its Euclidean magnitude √(Σ r[j]²). The set of all such combinations as the coefficients vary is the span of the input vectors.
The number of coefficients must equal the number of vectors, and all vectors must share the same dimension; otherwise the result is undefined. For 2D inputs the plot draws each input vector and the resulting vector from the origin. Inputs are unitless, so the result carries whatever units the vector entries represent.
Frequently asked questions
It is the sum you get by scaling each vector by a number (its coefficient) and adding them together: c₁·v₁ + c₂·v₂ + ... + cₖ·vₖ. The result is another vector of the same dimension.
Yes. Vectors can only be added component by component, so every vector must have the same dimension, and the number of coefficients must match the number of vectors.
It is the Euclidean length of the resulting vector, √(Σ r[j]²), which measures how far the combined vector reaches from the origin.
Also known as
TG we-Calculate Editorial Team. (2026). Linear Combination of Vectors Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/linear-combination-calculator
TG we-Calculate Editorial Team. "Linear Combination of Vectors Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/linear-combination-calculator.
TG we-Calculate Editorial Team, "Linear Combination of Vectors Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/linear-combination-calculator
@misc{wecalculate_linear_combination_calculator, title = {Linear Combination of Vectors Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/linear-combination-calculator}}, year = {2026}, note = {TG we-Calculate} }
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