Combinação Linear

The linear combination calculator allows you to find the resulting vector from combining two 2D vectors given scalar coefficients α and β. It is useful for students and professionals working with linear algebra, analytic geometry, or physics, as it quickly provides the weighted sum vector. Simply enter the components of vectors u and v, along with the scalars, and the result is displayed immediately.

The calculation follows the formula c = αu + βv, where u = (u1, u2) and v = (v1, v2). We multiply each component of u by α, each component of v by β, and add the corresponding components: c1 = α*u1 + β*v1 and c2 = α*u2 + β*v2. The result is a new vector representing the linear combination. This operation is fundamental for understanding vector spaces and linear systems.

Use this calculator when you need to quickly check linear combinations in problems of linear dependence, linear transformations, or when solving systems of equations. It is especially useful in linear algebra classes to verify exercises or explore how different scalars alter the resulting vector. It can also be applied in computer graphics to manipulate vectors.

Caution: ensure vectors u and v have the same dimension (2D). The calculator does not validate whether the vectors are linearly independent; that depends on context. Verify that scalars α and β are real numbers. For 3D problems, this tool is not suitable. Always check if the result makes geometric sense.