What this calculator does
The cross product returns a vector perpendicular to both inputs, with a length equal to the area of the parallelogram they span. It exists only in three dimensions, which is why it behaves so differently from the dot product.
The unit vectors demonstrate it cleanly. The x unit vector crossed with the y unit vector gives exactly (0, 0, 1.000), which is the z unit vector, with a magnitude of 1.000 for the unit square they span.
The formula
Each component of the result is a 2x2 determinant of the other two components of the inputs. The magnitude of the result is the area of the parallelogram the two vectors define.
| Term | Meaning |
|---|---|
| Cross product | A vector perpendicular to both inputs. |
| Parallelogram area | The magnitude of the cross product. |
| Right-hand rule | The convention determining which of the two perpendicular directions is chosen. |
The inputs explained
| Field | What to enter |
|---|---|
| a₁ (x) | First vector, x component. |
| a₂ (y) | First vector, y component. |
| a₃ (z) | First vector, z component. |
| b₁ (x) | Second vector, x component. |
| b₂ (y) | Second vector, y component. |
| b₃ (z) | Second vector, z component. |
When to use it
Finding a normal to a plane
Two vectors in a plane cross to give a vector perpendicular to it.
Calculating torque
Torque is the cross product of position and force.
Finding a triangle area in 3D
Half the cross product magnitude gives the area directly.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
What do different second vectors give?
The x unit vector crossed with three others.
| Second vector y component | Magnitude |a×b| | Triangle area (half) |
|---|---|---|
| (0, 0, 0) | 0 | 0 |
| (0, 1, 0) | 1.000 | 0.5000 |
| (0, 2, 0) | 2.000 | 1.000 |
Questions
Why does the cross product only work in 3D?
Because only in three dimensions is there a unique direction perpendicular to two given vectors. In two dimensions there is no room for it, and in four or more there is an entire plane of perpendicular directions rather than a single one.
What does a zero cross product mean?
That the two vectors are parallel or antiparallel, so they span no area. It is the counterpart of the dot product test: zero dot product means perpendicular, zero cross product means parallel.
Which perpendicular direction is chosen?
The one given by the right-hand rule: point your fingers along the first vector, curl them toward the second, and your thumb points along the result. Reversing the order reverses the direction.
Why is the magnitude an area?
Because it equals the product of the two lengths times the sine of the angle between them, which is exactly the formula for the area of a parallelogram with those sides.
For the scalar product instead, see the dot product calculator. For vector length and direction, see the vector magnitude calculator.