Calculator Inputs
Enter three 3D vectors. The page checks coplanarity through the scalar triple product and shows a geometric interpretation.
Formula used
Three vectors are coplanar when their scalar triple product is zero, or practically zero within a chosen tolerance.
For vectors A = (a1, a2, a3), B = (b1, b2, b3), C = (c1, c2, c3):
A · (B × C) = det | a1 a2 a3 |
| b1 b2 b3 |
| c1 c2 c3 |
If |A · (B × C)| ≤ tolerance, the vectors are treated as coplanar.
Volume of the parallelepiped = |A · (B × C)|
Why this works
The scalar triple product measures signed volume. A zero volume means the three directions collapse into a single plane.
Tolerance meaning
Small floating-point rounding can create tiny nonzero results. Tolerance helps classify nearly coplanar vectors more realistically.
How to use this calculator
- Enter the x, y, and z components for vectors A, B, and C.
- Choose a tolerance value to define how close to zero the determinant must be.
- Set the decimal precision for the displayed answers.
- Press Check Coplanarity to see the result above the form.
- Review the determinant, volume, cross products, plane equation, and graph.
- Use the CSV or PDF buttons to export the final calculation summary.
Example data table
| Case | Vector A | Vector B | Vector C | Determinant | Result |
|---|---|---|---|---|---|
| Planar basis | (1, 0, 0) | (0, 1, 0) | (1, 1, 0) | 0 | Coplanar |
| Nonzero volume | (1, 2, 3) | (0, 1, 4) | (2, 0, 1) | 11 | Not coplanar |
| Parallel pair | (1, 2, 3) | (2, 4, 6) | (0, 1, 1) | 0 | Coplanar |
| All zero-plane | (3, -1, 2) | (6, -2, 4) | (9, -3, 6) | 0 | Coplanar |
FAQs
1) What does coplanar mean for vectors?
Coplanar vectors lie on one plane through the origin when treated as free vectors. Their scalar triple product becomes zero.
2) Why is the scalar triple product important?
It measures signed three-dimensional volume. A zero value means the vectors do not span full 3D space and remain planar.
3) Why does the calculator ask for tolerance?
Real calculations may include rounding noise. Tolerance lets you treat very small determinant values as effectively zero.
4) Can parallel vectors still be coplanar?
Yes. If two vectors are parallel, they still lie in many possible planes through the origin. A third vector can share one of those planes.
5) What does a nonzero determinant mean?
It means the vectors span a genuine three-dimensional parallelepiped with nonzero volume, so they are not coplanar.
6) Does vector order change the coplanarity decision?
Order may change the determinant sign, but not its absolute value. The final coplanar or non-coplanar decision stays the same.
7) Can I use point coordinates here?
Yes, after converting points into direction vectors, usually by subtracting one common reference point from the others.
8) What does the graph help me understand?
The graph shows the three directions in space and, when possible, the normal vector of the plane used for interpretation.