How this calculator works
The cross product is perpendicular to both input vectors and follows the right-hand rule. Its magnitude equals the parallelogram area formed by the vectors. Parallel inputs, including a zero vector, produce the zero vector. Reconstructing B is not unique: adding any multiple of A leaves A cross B unchanged. Recovery returns the perpendicular, minimum-length solution and the full one-parameter family.
Formula
A cross B = (Ay Bz - Az By, Az Bx - Ax Bz, Ax By - Ay Bx). If C = A cross B and C dot A = 0, B0 = (C cross A)/|A|^2 and B = B0 + t A.
Worked example
For A = (1, 0, 0) and B = (0, 1, 0), A cross B = (0, 0, 1). Its magnitude is 1, giving a triangle area of 0.5 square coordinate units.
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Frequently asked questions
What happens when I swap A and B?
The result changes sign: B cross A = -(A cross B).
Can I use 2D vectors?
Append a zero z component to each vector. The result then points along the z axis.
Why is recovery a family?
The component of B parallel to A contributes nothing to the cross product, so it cannot be determined from C.