Cross Product Calculator
Compute the 3D cross product a × b and the magnitude of the resulting vector.
The magnitude |a × b| equals the area of the parallelogram spanned by a and b. The vector a × b is perpendicular to both a and b, with direction given by the right-hand rule.
How it works
- 1Enter the x, y, z components of vector a and vector b.
- 2The cross product a × b is computed as (a_y·b_z − a_z·b_y, a_z·b_x − a_x·b_z, a_x·b_y − a_y·b_x).
- 3|a × b| is the Euclidean length of the result — equal to the area of the parallelogram formed by a and b.
Use cases
- Physics: finding the torque vector or the normal to a surface from two edge vectors.
- Computer graphics: computing face normals for lighting and shading calculations.
- Engineering: determining the moment of a force about a point in 3D space.
Frequently asked questions
What does the direction of a × b represent?
a × b points perpendicular to the plane containing a and b, with the sense given by the right-hand rule: curl your fingers from a toward b and your thumb points along a × b.
When is the cross product zero?
a × b is the zero vector when a and b are parallel (or anti-parallel), because the angle between them is 0° or 180° and sin(0°) = sin(180°) = 0.
Is the cross product commutative?
No — a × b = −(b × a). Swapping the two vectors reverses the direction of the result. The magnitude stays the same, but the sign of every component flips.
Related tools
See all →Calculate the area and perimeter of a rectangle from its width and height, with a labeled diagram.
Calculate the area and circumference of a circle from its radius, with a labeled diagram.
Calculate the area (Heron’s formula) and perimeter of a triangle from its three sides.
Calculate the area and perimeter of a trapezoid from its two parallel sides, height, and legs.