The cross product

The cross product of two vectors A and B in ℝ³ is a third vector A × B perpendicular to both A and B, with direction given by the right-hand rule and magnitude |A × B| = |A| · |B| · sin(θ), where θ is the angle between A and B. Unlike the dot product (which returns a number), the cross product returns a vector. Defined only in 3D (and in a special 7D case); generalisations for other dimensions require the exterior product (wedge product).

Component formula

If A = (a_x, a_y, a_z) and B = (b_x, b_y, b_z): A × B = (a_y·b_z − a_z·b_y, a_z·b_x − a_x·b_z, a_x·b_y − a_y·b_x). Mnemonic: formal 3×3 determinant with basis ijk on the top row. Example: A = (1, 2, 3), B = (4, 5, 6) → A × B = (2·6 − 3·5, 3·4 − 1·6, 1·5 − 2·4) = (−3, 6, −3). Check: (−3, 6, −3) · (1, 2, 3) = 0 (⊥ A); · (4, 5, 6) = 0 (⊥ B). ✓

Geometric interpretation

The magnitude of the cross product equals the parallelogram area formed by the two vectors: |A × B| = |A|·|B|·sin(θ). Parallel vectors (θ = 0° or 180°) have zero cross product (zero area). Orthogonal vectors (θ = 90°) have maximum magnitude |A|·|B|. The direction is perpendicular to the (A, B) plane, with orientation following the right-hand rule: point the index finger along A and middle finger along B; the thumb points along A × B.

Algebraic properties

• Anticommutative: A × B = −B × A (swapping order reverses direction).
• Distributive: A × (B + C) = A × B + A × C.
• Non-associative: (A × B) × C ≠ A × (B × C) in general.
• Jacobi identity: A × (B × C) + B × (C × A) + C × (A × B) = 0.
• BAC-CAB rule: A × (B × C) = B(A·C) − C(A·B).
• A × A = 0 for any A.

Triple product and volume

The triple product A · (B × C) is a scalar equal to the determinant of the 3×3 matrix with A, B, C as rows. Its absolute value is the parallelepiped volume formed by the three vectors: V = |A · (B × C)|. If the triple product is zero, the vectors are coplanar (linearly dependent). The sign indicates orientation (right-handed if positive). Cyclic: A·(B×C) = B·(C×A) = C·(A×B).

Physics applications

The cross product describes many physical quantities requiring perpendicular directionality: angular momentum L = r × p; torque τ = r × F; Lorentz force F = q·v × B (charged particle in magnetic field); Poynting vector S = E × H (EM energy flux); rotational velocity v = ω × r (ω angular velocity); Coriolis acceleration a_C = −2·ω × v. The right-hand rule is ubiquitous in electromagnetism and rotational mechanics.

Graphics and engineering

In 3D computer graphics: compute surface normals (N = (V₂−V₁) × (V₃−V₁) for a triangle), basis for lighting calculations and backface culling; build local coordinate frames (tangent-bitangent-normal for normal mapping). In robotics: articulated-robot joint computations (inverse kinematics). In navigation: determining normals to orbital planes, course calculations. In CAD: polygon face areas, tetrahedral volumes (one-sixth of triple product).

2D and 7D variants

In 2D the cross product reduces to a scalar representing the z-component of the 3D vector if 2D vectors are embedded in the xy-plane: A × B = a_x·b_y − a_y·b_x. Useful for triangle areas, polygon orientation, left/right point-line tests. A 7D cross product exists via octonions. In dimensions other than 3 and 7 there's no bilinear cross product; instead the wedge product returns a bivector.

How to use

Four modes: basic cross product (A, B 3D → A×B, |A×B|, angle, parallelogram area); triple product (A, B, C → A·(B×C), parallelepiped volume); parallelogram area (A, B → area, diagonals); tetrahedral volume (A, B, C → V = |A·(B×C)|/6). Enter vectors as comma-separated lists of 3 components. The calculator flags parallel vectors (zero cross product) and coplanar vectors (zero triple product).

Updated July 2026