Cross Product Calculator (3D)
Written by Thierno Sadou Diallo, formula verified per our methodology • Last checked on 9/6/2026
The cross product of two vectors u and v is calculated component by component: (u₂v₃−u₃v₂, u₃v₁−u₁v₃, u₁v₂−u₂v₁). The cross product of (2,3,4) and (5,6,7) gives the vector (−3, 6, −3).
Explanation
Unlike the dot product, which gives a plain number, the cross product of two 3D vectors produces a third vector: perpendicular to both starting vectors, with a magnitude equal to the area of the parallelogram they form, and a direction following the right-hand rule (or corkscrew rule). This operation only exists for 3-dimensional vectors (or 7-dimensional, a far rarer case) — there's no direct equivalent in 2D, where the determinant instead plays an analogous role in measuring a signed area. The cross product is used in physics to calculate a torque (moment of force, closely tied to the forces involved — see our Newton's second law force calculator), the Lorentz force on a moving charge in a magnetic field, or in 3D computer graphics to find the normal to a surface from two vectors lying in that surface. An important property to know: unlike ordinary multiplication, the cross product is not commutative — swapping the order of the two vectors flips the sign of the result (u×v = −(v×u)).
Example: cross product of (2,3,4) and (5,6,7)
Inputs
Vector u = (2, 3, 4). Vector v = (5, 6, 7).
Calculation
x = 3×7 − 4×6 = 21 − 24 = −3. y = 4×5 − 2×7 = 20 − 14 = 6. z = 2×6 − 3×5 = 12 − 15 = −3.
Result
The cross product of these two vectors is (−3, 6, −3).
Frequently asked questions
Why is the result perpendicular to both starting vectors?
This is a property that follows directly from the definition of the cross product: you can verify that the dot product of the result with u, and separately with v, is always zero — the very definition of perpendicularity between two vectors. This property is what makes the cross product useful for finding a normal to a plane.
What happens if the two vectors are parallel?
The cross product of two parallel (or collinear) vectors is always the zero vector (0, 0, 0), because the area of the parallelogram they would form is zero — two aligned vectors bound no surface at all. This is a quick, commonly used test to check whether two vectors are collinear.
Does the order of the vectors matter?
Yes, unlike the dot product (commutative), the cross product is not: u×v and v×u have the same magnitude but opposite direction (u×v = −v×u). Swapping the order of the two vectors in the form therefore flips the sign of all three components of the result.