Cross Product Calculator
Calculate the cross product of two 3D vectors.
About Cross Product Calculator
The cross product of two 3D vectors produces a third vector perpendicular to both, with magnitude equal to the area of the parallelogram they span. It is used in computer graphics, physics simulations, and 3D geometry calculations within ML pipelines.
Formula
Cross Product
a × b = (a_y*b_z - a_z*b_y, a_z*b_x - a_x*b_z, a_x*b_y - a_y*b_x)
- a, b
- 3-dimensional input vectors
Worked example
With the default values below, Cross Product Calculator returns Magnitude of 7.348469. Change any input above and every figure here is recalculated in your browser.
Inputs used
- Vector A (3D)
- 1, 2, 3
- Vector B (3D)
- 4, 5, 6
Results
- Magnitude
- 7.348469
- X Component
- -3.000000
- Y Component
- 6.000000
- Z Component
- -3.000000
Step by step
- 1x component: a_y×b_z − a_z×b_y= -3.000000
- 2y component: a_z×b_x − a_x×b_z= 6.000000
- 3z component: a_x×b_y − a_y×b_x= -3.000000
- 4Magnitude= 7.348469
Assumptions and limits
- The result is derived only from the 2 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 6 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
Why is the cross product only defined in 3D?
The cross product as a vector operation is specific to 3D (and 7D) because it relies on the structure of the space to produce a unique perpendicular vector. In other dimensions, the wedge product generalizes this concept.
How is the cross product used in ML?
It is used in 3D point cloud processing, computer vision for surface normal estimation, robotics, and any ML application involving 3D geometric reasoning.