Free · Coordinate geometry calculators 📈

3D Cross Product and Vector Components

Compute a three-dimensional cross product from signed Cartesian components, with zero-vector handling, conditional vector summaries and a worked example.

  • Formula & worked example
  • Private in your browser
  • No signup
Go to tool

Calculator inputs

Loading calculator…

How to use this calculator

  1. Enter the known values in the units shown. Results update as you type.
  2. Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
  3. Use the worked example to check the method. Reset restores the starting fields.

Private by default

Inputs and results stay in this browser tab. Bookify does not upload or store the values you enter.

Formula and method

For A=(1,2,3) and B=(4,5,6), the cross product A×B is C=(−3,6,−3). Reversing the order reverses the result.

C_x=A_yB_z−A_zB_y; C_y=A_zB_x−A_xB_z; C_z=A_xB_y−A_yB_x.

Worked example

Enter these known values and leave the other values blank.

A x-component
1
A y-component
2
A z-component
3
B x-component
4
B y-component
5
B z-component
6
C x-component
-3
C y-component
6
C z-component
-3

Assumptions and limitations

  • Components use a common right-handed orthonormal Cartesian basis. Negative and zero components are valid. The order is A×B, not B×A.
  • The result is perpendicular to both input vectors. Its magnitude equals |A||B|sin(θ); its direction follows the right-hand rule. Parallel, antiparallel and zero vectors give a zero result.
  • The six input components and three read-only result components retain the captured interface. Enter all components of A and B for a complete forward result. The underlying scalar formula graph also retains component inverses when enough compatible independent quantities are known; a zero coefficient does not determine a unique missing component.
  • A whole input vector cannot in general be recovered uniquely from only the other vector and their cross product: adding a component parallel to that known vector leaves the product unchanged. Supply enough independent component information.
  • Use consistent component units within each vector; the output has the product of those units. These bare numeric controls do not convert units or transform between coordinate bases.
  • Overdetermined inconsistent component values require correction before using the result. Blank means unknown; enter zero explicitly when an input component is zero.

Common questions

What happens when the vectors are parallel?

Their cross product is the zero vector, including antiparallel inputs. A zero product by itself does not identify a unique missing input vector.

Does changing vector order matter?

Yes. B×A=−(A×B). Each result component changes sign when you exchange the two inputs.

Why can a missing component remain blank?

If its coefficient is zero in every applicable equation, those equations do not determine it uniquely. Enter more independent information instead of dividing by zero.

References

Cross-product component equations hold for zero and signed real components. Twelve captured component inverses divide by a vector component; guard only those division variants when the coefficient is zero. This leaves a non-unique unknown blank and permits another nonzero-coefficient equation to solve it. Forward products and differences retain valid zero vectors and parallel vectors.

Explore coordinate geometry calculators 📈 or browse all math calculators.