Produto Vetorial
- Created by
- Renato Passos, Eng. de Software
- Reviewed by
- Renato Passos, Eng. de Software
Last updated: Apr 18, 2026
Formula
(u×v) = (u₂v₃−u₃v₂, u₃v₁−u₁v₃, u₁v₂−u₂v₁)
About this calculator
The cross product calculator computes the vector orthogonal to two vectors u and v in three-dimensional space (R³). The result is a vector perpendicular to the plane formed by u and v, with magnitude equal to the area of the parallelogram they define. This operation is fundamental in linear algebra, geometry, and physics.
The calculation follows the formula u × v = (u₂v₃ − u₃v₂, u₃v₁ − u₁v₃, u₁v₂ − u₂v₁). Simply enter the coordinates of the two vectors and the tool automatically performs the products and subtractions, returning the resulting vector. The result can be used to determine surface normals, torques, or moments.
Use this calculator whenever you need to find a vector perpendicular to two given vectors, for example, when calculating the normal of a triangle in computer graphics, determining the direction of a magnetic force (F = q v × B), or computing torque τ = r × F. It is also useful in analytic geometry problems to verify orthogonality.
Caveats: ensure the vectors are not parallel (otherwise the cross product will be zero). Order matters: u × v = − (v × u). Check that coordinates are in the correct format (x, y, z). The calculator assumes vectors are in R³; for R², consider the third coordinate as zero.
Frequently asked questions
What does it mean if the cross product is zero?
It means the vectors are parallel or one of them is the zero vector. In that case, there is no unique perpendicular vector.
Can I use this calculator for 2D vectors?
Yes, just enter the third coordinate as zero. The result will be a vector perpendicular to the xy plane.
Does the order of vectors affect the result?
Yes, u × v is the opposite of v × u. The magnitude is the same, but the direction is reversed.
How can I verify the result is correct?
Check that the dot product of the result with u and with v is zero (orthogonality). You can use our dot product calculator.
What is the unit of the result?
The unit is the product of the units of the input vectors. For example, if u and v are in meters, the result is in square meters.