Cross Product Calculator: Complete Guide to 3D Vector Product, Determinant Method, and Applications
The cross product (vector product) is an operation on two 3D vectors that produces a third vector perpendicular to both inputs. Written A × B, its magnitude equals the area of the parallelogram formed by A and B, and its direction follows the right-hand rule. The cross product is essential in physics (torque, magnetic force, angular momentum), computer graphics (surface normals, lighting), and engineering (structural analysis, electromagnetic theory). Unlike the dot product which yields a scalar, the cross product yields a vector — making it the complementary operation for complete 3D vector analysis.
The Determinant Formula
The cross product is computed as a 3×3 determinant: A × B = |[î, ĵ, k̂], [a₁, a₂, a₃], [b₁, b₂, b₃]|. Expanding along the first row: A × B = (a₂b₃ − a₃b₂)î − (a₁b₃ − a₃b₁)ĵ + (a₁b₂ − a₂b₁)k̂. Each component is a 2×2 determinant (cofactor expansion). This formula is elegant because it unifies the algebraic computation with the geometric meaning: the determinant naturally encodes both the perpendicularity and the area-magnitude relationship. Our calculator shows this determinant visually with the matrix and cofactor expansion.
= (a₂b₃-a₃b₂)î - (a₁b₃-a₃b₁)ĵ + (a₁b₂-a₂b₁)k̂
|A × B| = |A||B|sinθ = parallelogram area
Properties:
A × B = -(B × A) (anti-commutative)
A × A = 0 (self-cross = zero)
A × B ⊥ A and A × B ⊥ B
Geometric Meaning
The cross product has profound geometric significance. Its magnitude |A × B| = |A||B|sinθ equals the area of the parallelogram spanned by A and B. Half this value gives the area of the triangle formed by the two vectors. Its direction is perpendicular to the plane containing A and B, determined by the right-hand rule: point your fingers along A, curl them toward B, and your thumb points in the direction of A × B. This perpendicularity is what makes cross products essential for computing surface normals in 3D graphics.
Cross Product in Physics
Torque: τ = r × F gives the rotational effect of a force F applied at position r from the pivot. The magnitude is |r||F|sinθ, and the direction is the axis of rotation. Magnetic force: F = qv × B gives the force on a charged particle moving through a magnetic field. This is why charged particles spiral in magnetic fields — the force is always perpendicular to velocity. Angular momentum: L = r × p = r × mv. Electromagnetic induction: Faraday’s law uses the cross product to describe how changing magnetic fields create electric fields.
Cross Product in Computer Graphics
Surface normals, essential for lighting calculations, are computed via cross products. Given two edges of a triangle (vectors along the edges), their cross product gives the surface normal direction. This normal determines how light reflects off the surface (Lambert’s law: brightness proportional to N·L where N is normal, L is light direction). The cross product also determines triangle winding order (clockwise vs counter-clockwise) for backface culling, computes tangent and bitangent vectors for normal mapping, and calculates the area of 3D triangles for physics simulation and collision detection.
Properties and Triple Products
The cross product is anti-commutative: A × B = -(B × A). It is not associative: (A × B) × C ≠ A × (B × C) in general. The scalar triple product A · (B × C) gives the volume of the parallelepiped formed by three vectors. If this volume is zero, the vectors are coplanar. The vector triple product A × (B × C) = B(A·C) − C(A·B) is known as the BAC-CAB rule and is used extensively in electromagnetic theory and classical mechanics.
How to Use This Calculator
Enter the three components (x, y, z) for Vector A and Vector B. The calculator computes A × B using the determinant method, showing the 3×3 determinant matrix with cofactor expansion. The 3D projection Canvas shows all three vectors with color coding: blue for A, green for B, red for the cross product result (perpendicular to both). Stats include the result components, magnitude (parallelogram area), and sinθ. The step-by-step solution walks through each component calculation with the 2×2 sub-determinants.