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Coplanar Vectors Calculator

Work out coplanar vectors instantly with clear inputs, formula shown and shareable results.

About Coplanar Vectors Calculator

Three vectors are coplanar exactly when the parallelepiped they span has zero volume, so the test is whether the scalar triple product a·(b × c) vanishes.

Formula

Coplanarity test

a · (b × c) = 0 ⇔ the three vectors are coplanar

Worked example

With the default values below, Coplanar Vectors Calculator returns Result of The three vectors are coplanar. Change any input above and every figure here is recalculated in your browser.

Inputs used

First vector
1, 2, 3
Second vector
2, 3, 4
Third vector
3, 5, 7

Results

Result
The three vectors are coplanar
Scalar triple product
0.00000000
Volume spanned
0.00000000
Normal to the plane of the first two
(-1, 2, -1)

Step by step

  1. 1Values used
  2. 2Coplanarity test
  3. 3Result= The three vectors are coplanar
  4. 4Scalar triple product= 0.00000000
  5. 5Volume spanned= 0.00000000
  6. 6Normal to the plane of the first two= (-1, 2, -1)

Assumptions and limits

  • The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
  • In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
  • Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

Are (1,2,3), (2,3,4) and (3,5,7) coplanar?

Yes — the third is the sum of the first two, so the triple product is 0.

Are any two vectors always coplanar?

Yes. Two vectors from a common origin always determine a plane, unless one is zero.

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