Vector Cross Product Calculator
A New Vector, Perpendicular to Both
Where the dot product collapses two vectors into a scalar, the cross product does the opposite: it builds a brand-new vector, one that stands perpendicular to both original 3D vectors at once. Its length also encodes something useful — the area of the parallelogram the two original vectors span.
The Formula
A × B = (Ay·Bz−Az·By, Az·Bx−Ax·Bz, Ax·By−Ay·Bx)
Worked Examples
| Vector A | Vector B | A × B |
|---|---|---|
| (1, 0, 0) | (0, 1, 0) | (0, 0, 1) |
| (2, 3, 4) | (5, 6, 7) | (−3, 6, −3) |
The first row is the standard basis result: the x-axis unit vector crossed with the y-axis unit vector gives the z-axis unit vector, the foundation of the right-hand rule.
Where This Matters
- Surface normals in 3D graphics — the cross product of two edge vectors of a triangle gives the direction the surface faces.
- Torque in physics — torque equals the cross product of a position vector and a force vector.
- Parallelogram and triangle area — the magnitude of the cross product directly gives the area spanned by two vectors, without needing angles.
How to Use This Calculator
- Enter Vector A's x, y, and z components.
- Enter Vector B's x, y, and z components.
- Select Calculate to get the resulting perpendicular vector.
Related Calculations
Need a scalar result instead? Use the Vector Dot Product Calculator, or measure the angle between the two vectors with the Angle Between Vectors Calculator.