Dependência Linear (2 vetores)
- Created by
- Renato Passos, Eng. de Software
- Reviewed by
- Renato Passos, Eng. de Software
Last updated: Apr 18, 2026
Formula
det = 0 ⇒ LD
About this calculator
This calculator checks if two vectors in the plane or space are linearly dependent (LD) or independent (LI) using the determinant of the matrix formed by them. If the determinant is zero, the vectors are LD; otherwise, they are LI. The tool accepts two-dimensional or three-dimensional vectors with real coordinates.
The calculation is simple: enter the coordinates of vectors u and v. For 2D vectors, the determinant is u₁·v₂ - u₂·v₁. For 3D vectors, compute the determinant of the 2x3 matrix (two rows). If the result is zero (within a tolerance for numerical errors), the vectors are dependent; otherwise, independent.
Use this calculator in linear algebra, analytic geometry, or physics problems to know if vectors represent the same direction (LD) or different directions (LI). For example, checking if forces act along the same line, or if points are collinear. It is also useful to determine if a set of vectors forms a basis.
Cautions: the calculator only considers two vectors. For more vectors, linear dependence requires rank verification. Also, zero vectors (0,0) are always LD with any vector. The tolerance for zero is 1e-9, so very small values may be considered zero. Always interpret the result in the context of the problem.
Frequently asked questions
What does it mean for two vectors to be linearly dependent?
It means one vector can be written as a linear combination of the other, i.e., they are parallel (or one is zero). Geometrically, they lie on the same line.
What if the determinant is a very small number, like 1e-12?
The calculator considers it zero if the absolute value is less than 1e-9. So 1e-12 will be considered zero and the vectors will be classified as LD.
Can I use it for vectors with more than 3 coordinates?
No. This calculator is only for 2D or 3D vectors. For higher dimensions, another method (like row reduction) is needed.
What is the difference between linear dependence and independence?
Linear dependence means the vectors are not linearly independent, i.e., there is a nontrivial linear combination that gives the zero vector. For two vectors, this is equivalent to being parallel.
What if I enter non-numeric coordinates?
The calculator expects numbers. If you type letters or symbols, an error will occur. Make sure to use only real numbers.