HesapHanem

Linear Independence Calculator

Enter the components of two 3D vectors to check linear independence.

How it is calculated

The tool tests two vectors in three dimensions, u = (a, b, c) and v = (d, e, f). It computes the cross product u × v = (b·f − c·e, c·d − a·f, a·e − b·d) and its length √(x² + y² + z²), which equals the area of the parallelogram the two vectors span. The vectors are linearly independent when that length is clearly above zero, and dependent (parallel, or one of them the zero vector) when it is zero.

Because of floating-point rounding, "zero" is judged relative to the size of the vectors: the cross product counts as zero when its length is below one billionth of the product of the two vector lengths. So decimal inputs that are exact multiples, such as (0.1, 0.2, 0.3) and (0.3, 0.6, 0.9), are correctly reported as dependent. When dependent, the cross product is displayed as (0.00, 0.00, 0.00). All six components are required, and only two vectors are tested, not larger sets.

Worked example

Take u = (1, 2, 3) and v = (4, 5, 6). First component: 2·6 − 3·5 = 12 − 15 = −3. Second: 3·4 − 1·6 = 12 − 6 = 6. Third: 1·5 − 2·4 = 5 − 8 = −3. The cross product is (-3.00, 6.00, -3.00), which is not zero, so the vectors are independent. Its length is √(9 + 36 + 9) = √54 = 7.35. Replacing v with (2, 4, 6), which is 2u, gives (0.00, 0.00, 0.00) and the result Dependent.

Try this example →

Common mistakes

  • Entering the six numbers in the wrong order. The first three (a, b, c) are the first vector and the last three (d, e, f) are the second one; mixing coordinates across vectors changes the answer.
  • Expecting a nearly parallel pair to look non-zero. For (1, 2, 3) and (2, 4, 6.001) the verdict is Independent, but the cross product (0.002, −0.001, 0) is displayed as (0.00, 0.00, 0.00) because the output is rounded to two decimals.
  • Using the tool for three or more vectors or for vectors in other dimensions. It tests exactly two 3D vectors; a third vector needs a determinant or rank test.
  • Forgetting that a zero vector is always dependent. If one vector is (0, 0, 0), the cross product is zero and the tool reports Dependent.
  • Reading the cross product length as a measure of how independent the vectors are. It depends on the vector lengths as well as the angle between them.

About this calculator

Two vectors in three-dimensional space are linearly independent when neither is a scalar multiple of the other — geometrically, when they don't point along the same line. The cleanest way to test this in 3D is the cross product: if u × v comes out as the zero vector, the two vectors are parallel (dependent); if it's non-zero, they span a plane and are independent.

This calculator computes that cross product directly from the vectors' components and reads off independence from the result. The concept underlies core linear algebra topics — determining whether a set of vectors can form a basis, whether a matrix is invertible, and whether a system of equations has a unique solution — so it comes up constantly for students working through vector spaces and in applied fields like computer graphics and physics where independent direction vectors are needed to define a coordinate frame.

Was this helpful?

You can pick up to 3 reactions.

Comments (1)

  • Editorial team

    A handy use beyond textbook exercises: checking whether three points sit on one straight line. Subtract the first point from the other two to get two vectors, enter them, and a Dependent verdict means the points are collinear. Because the test is relative to vector size, measured coordinates with a little noise will usually come out Independent, so judge near-zero results by the cross product length instead. Do you have a geometry or graphics problem where you need to test more than two vectors?

Popular calculators

Loan Calculator

Monthly payment, total interest and full amortization schedule.

Percentage Calculator

Percent change, percent of, and ratio calculations.

Discount Calculator

Calculate final price, savings, and discount amount.

BMI Calculator

Body mass index from height and weight (metric or imperial).

Age Calculator

Compute your exact age in years, months and days from a birth date.

Compound Interest Calculator

Calculate the final amount and interest earned with compounding.

All Calculators