Linear Independence Calculator — 3 Vectors in ℝ³
Enter three vectors v₁, v₂, v₃ in ℝ³ to determine whether they are linearly independent. The calculator forms the 3×3 matrix [v₁ | v₂ | v₃] and tests whether its determinant is non-zero.
det ≠ 0 — vectors are LINEARLY INDEPENDENT
- 1
M₁₁ = y₂z₃ − z₂y₃
1 × 1 − 0 × 0 = 1 - 2
M₁₂ = x₂z₃ − z₂x₃
0 × 1 − 0 × 0 = 0 - 3
M₁₃ = x₂y₃ − y₂x₃
0 × 0 − 1 × 0 = 0 - 4
det = x₁M₁₁ − y₁M₁₂ + z₁M₁₃
1 × 1 − 0 × 0 + 0 × 0 = 1Cofactor expansion along the first row. - 5
|det|
|1| = 1
How does this calculator work?
Stack v₁, v₂, v₃ as columns of a 3×3 matrix and compute the determinant. det ≠ 0 → linearly independent (they form a basis for ℝ³). det = 0 → linearly dependent (they are coplanar and do not span ℝ³). The determinant equals the signed volume of the parallelepiped the three vectors define.
Formula
How this is calculated
Three vectors in ℝ³ are linearly independent if the only scalars c₁, c₂, c₃ satisfying c₁v₁ + c₂v₂ + c₃v₃ = 0 are c₁ = c₂ = c₃ = 0. A compact test is to stack the vectors as columns of a 3×3 matrix and compute its determinant by cofactor expansion along the first row: det = x₁(y₂z₃ − y₃z₂) − y₁(x₂z₃ − x₃z₂) + z₁(x₂y₃ − x₃y₂). A non-zero determinant means the three vectors form a basis for ℝ³ and span the entire three-dimensional space. A zero determinant means at least one vector is a linear combination of the other two and the span is at most two-dimensional (a plane or a line).
Geometrically, the determinant equals the signed volume of the parallelepiped spanned by the three vectors. Zero volume means the parallelepiped is flat — all three vectors lie in a common plane (they are coplanar), which is exactly the definition of linear dependence in ℝ³.
The calculator uses a tolerance of |det| < 1e-10 to flag near-zero determinants as dependent, handling rounding errors for inputs that are theoretically dependent. For exact integer or rational inputs the floating-point result is accurate to at least 10 significant figures.
Frequently asked questions
This calculator tests exactly three vectors in ℝ³. For two vectors in ℝ² the independence test is |x₁y₂ − x₂y₁| ≠ 0. For two vectors in ℝ³ they are independent unless one is a scalar multiple of the other (cross product is zero). For four or more vectors in ℝ³ they are always dependent by the rank theorem.
Det = 0 means the three vectors are coplanar — they all lie in some plane through the origin. Their span is a 2D subspace (a plane) or lower, never all of ℝ³. The matrix rank is at most 2. You can find the actual span dimension by row-reducing the matrix.
Swapping two columns flips the sign of the determinant but not its magnitude, so the independence test (|det| ≠ 0) gives the same result regardless of the order you list the three vectors.
Also known as
TG we-Calculate Editorial Team. (2026). Linear Independence Calculator — 3 Vectors in ℝ³ [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/linear-independence-calculator
TG we-Calculate Editorial Team. "Linear Independence Calculator — 3 Vectors in ℝ³." TG we-Calculate. 2026. https://we-calculate.com/calculator/linear-independence-calculator.
TG we-Calculate Editorial Team, "Linear Independence Calculator — 3 Vectors in ℝ³," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/linear-independence-calculator
@misc{wecalculate_linear_independence_calculator, title = {Linear Independence Calculator — 3 Vectors in ℝ³}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/linear-independence-calculator}}, year = {2026}, note = {TG we-Calculate} }
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