The cross product takes two 3D vectors, a and b, and returns a new vector perpendicular to both — a building block for finding surface normals, torque, angular momentum, and the area of a parallelogram or triangle in 3D space. This calculator computes a × b, its magnitude, and the parallelogram area it represents, all from the same two input vectors.
How the Cross Product Calculator works
The calculator applies the determinant-expansion formula a × b = (a2b3 − a3b2, a3b1 − a1b3, a1b2 − a2b1) directly to the six components you enter. The result is itself a 3D vector, not a single number — its direction follows the right-hand rule, and it's always perpendicular to both a and b.
The magnitude of that result vector, |a × b|, doubles as the area of the parallelogram formed by placing a and b tail-to-tail. The Cross Product, Magnitude, and Parallelogram Area tabs are three different views of the exact same underlying calculation.
Inputs and what they mean
Vector a and vector b are entered as their x, y, and z components. Unlike a general-purpose vector calculator, there's no 2D mode here — the cross product in its full vector form only exists in 3D, so all six fields are required before any tab computes a result.
Limits and edge cases
Whenever a and b are parallel (including the case where one of them is the zero vector), the cross product is the zero vector and the parallelogram area is 0 — there's no twist between two parallel directions for the cross product to capture. The calculator flags this case explicitly rather than showing a misleadingly precise near-zero number.
This calculator works with any consistent unit for the vector components — just use the same unit for every component you enter, since the output units multiply (e.g., meters × meters = square meters for the area).