Cross Product Calculator
Calculate the 3D cross product, its magnitude and the angle between two vectors
A × B
How we calculated it
About the Cross Product Calculator
The cross product of two three-dimensional vectors produces a third vector perpendicular to both. Its magnitude represents the area of the parallelogram spanned by the two input vectors and its direction follows the right-hand rule.
This operation is central in torque, rotational mechanics and three-dimensional geometry. Unlike the dot product, order matters because reversing the inputs reverses the resulting vector.
How to Use the Cross Product Calculator
- Enter the x, y and z components of both vectors.
- Calculate the cross product in the intended order a×b.
- Review the resulting components and magnitude.
- Use the right-hand rule to interpret direction when the vector represents a physical quantity.
How the Calculation Works
For a=(a₁,a₂,a₃) and b=(b₁,b₂,b₃), a×b=(a₂b₃−a₃b₂, a₃b₁−a₁b₃, a₁b₂−a₂b₁). Each output component is formed from a signed two-by-two determinant of the other coordinate components.
The magnitude is |a×b|=|a||b|sinθ, so parallel vectors have a zero cross product while perpendicular vectors produce the maximum magnitude for fixed vector lengths. Reversing the input order changes the sign of every component: b×a=−(a×b).
Worked example
Cross Product Calculator Worked Example
(1,0,0) × (0,1,0)
The result is (0,0,1), the positive z-direction under a standard right-handed coordinate system.
Result guide
Understanding Your Cross Product Calculator Result
The resulting vector is perpendicular to the plane containing the inputs. Reversing the order gives the opposite direction: b×a=−(a×b). Its magnitude can also be interpreted geometrically as twice the area of the triangle formed by the two input vectors from a common origin. That makes the operation useful in 3D area calculations as well as in mechanics.
Before you rely on the answer
Assumptions and Limitations
- Vectors are three-dimensional under the standard cross-product definition.
- A right-handed coordinate convention is assumed for direction.
- Inputs use the same coordinate basis.
Practical tip
Using This Result in Practice
When using the cross product for torque, order the vectors consistently as position vector × force vector. Swapping them changes the sign and therefore the torque direction.
Related calculations
Useful Calculations to Continue With
If you need scalar alignment or an angle relationship instead, use the Dot Product Calculator.
If the cross product is being used specifically for a turning moment, use the Torque Calculator for the physical interpretation.
Common questions
Cross Product Calculator FAQ
What direction does a cross product point?
Perpendicular to both input vectors according to the right-hand rule.
Why does order matter?
Reversing the vector order reverses the direction of the result.
What does a zero cross product mean?
The vectors are parallel, anti-parallel or one vector has zero magnitude.