Vector Cross Product Calculator
Reviewed by Hamza Gujjar, Algorithm Architect & Software Developer
Last updated: August 19, 2026A New Vector, Perpendicular to Both
Where the dot product collapses two vectors into a scalar, the cross product does the opposite: it builds a brand-new vector, one that stands perpendicular to both original 3D vectors at once. Its length also encodes something useful — the area of the parallelogram the two original vectors span.
The Formula
Worked Examples
| Vector A | Vector B | A × B |
|---|---|---|
| (1, 0, 0) | (0, 1, 0) | (0, 0, 1) |
| (2, 3, 4) | (5, 6, 7) | (−3, 6, −3) |
The first row is the standard basis result: the x-axis unit vector crossed with the y-axis unit vector gives the z-axis unit vector, the foundation of the right-hand rule.
Where This Matters
- Surface normals in 3D graphics — the cross product of two edge vectors of a triangle gives the direction the surface faces.
- Torque in physics — torque equals the cross product of a position vector and a force vector.
- Parallelogram and triangle area — the magnitude of the cross product directly gives the area spanned by two vectors, without needing angles.
How to Use This Calculator
- Enter Vector A's x, y, and z components.
- Enter Vector B's x, y, and z components.
- Select Calculate to get the resulting perpendicular vector.
Related Calculations
Need a scalar result instead? Use the Vector Dot Product Calculator, or measure the angle between the two vectors with the Angle Between Vectors Calculator.
Principles of the Vector Cross Product (Vector Product)
The vector cross product — denoted by the symbol × — is a binary operation defined strictly in three-dimensional Euclidean space (ℝ³) that takes two 3D vectors u and v and produces a new vector w = u × v that is strictly perpendicular (orthogonal) to both input vectors, with its directional sense dictated by the Right-Hand Rule.
The Matrix Determinant Formulation
The cross product of vectors u = 〈ux, uy, uz〉 and v = 〈vx, vy, vz〉 is computed by evaluating a formal 3×3 matrix determinant using standard unit basis vectors i, j, k:
Geometric Area and Magnitude
The scalar magnitude of the cross product equals the physical surface area of the parallelogram spanned by the two vectors:
- Parallel / Collinear Vectors (u × v = 0): When vectors are parallel or anti-parallel (θ = 0° or 180°), sin(θ) = 0, producing a null zero vector.
- Triangle Area: The area of a triangle formed by two vectors equals exactly half the cross product magnitude: Areatriangle = 0.5 × ||u × v||.
Fundamental Algebraic Properties
- Anti-Commutativity (Non-Commutative): Reversing the order reverses vector direction: u × v = -(v × u).
- Distributivity: u × (v + w) = (u × v) + (u × w).
- Scalar Triple Product (Volume of Parallelepiped): Volume = | u · (v × w) |.
Step-by-Step Worked Calculation Example
Example: Calculating Mechanical Torque on a Rotating Robotic Arm
Problem: A robotic actuator applies a 3D force vector F = 〈20.0, -40.0, 50.0〉 Newtons at position lever arm r = 〈0.50, 1.20, 0.00〉 meters from the rotational pivot joint. Calculate: (1) The mechanical torque vector τ = r × F in Newton-meters (N·m); and (2) The total torque magnitude ||τ||.
Step 1: Compute cross product components:
τx = (ry × Fz) - (rz × Fy) = (1.20 × 50.0) - (0.0 × -40.0) = 60.0 - 0.0 = +60.00 N·m
τy = (rz × Fx) - (rx × Fz) = (0.0 × 20.0) - (0.50 × 50.0) = 0.0 - 25.0 = -25.00 N·m
τz = (rx × Fy) - (ry × Fx) = (0.50 × -40.0) - (1.20 × 20.0) = -20.0 - 24.0 = -44.00 N·m
Torque Vector τ = 〈60.0, -25.0, -44.0〉 N·m
Step 2: Calculate scalar torque magnitude:
||τ|| = √[ (60.0)² + (-25.0)² + (-44.0)² ] = √[ 3,600 + 625 + 1,936 ] = √6,161 = 78.49 N·m
Conclusion: The joint experiences a net rotational torque of 78.49 N·m.
Applications in Electromagnetics: Lorentz Magnetic Force
In electrodynamics, a charged particle q moving with velocity v through magnetic field B experiences magnetic force: F = q × (v × B), bending electron beams in particle accelerators.
The Vector Triple Product Expansion (BAC-CAB Identity)
In theoretical mechanics and vector calculus, evaluating repeated cross products involves the fundamental Vector Triple Product Identity (BAC-CAB Rule):
This identity allows physicists to reduce complex non-associative cross products into straightforward dot products and scalar vector scalings.
Jacobi Identity and Lie Algebra Structure
Because the cross product is non-associative (a × (b × c) ≠ (a × b) × c), it satisfies the Jacobi Identity:
Equipped with anti-commutativity and the Jacobi identity, three-dimensional Euclidean space under the cross product operation forms the Lie algebra so(3), which governs 3D spatial rotations in quantum angular momentum and aerospace spacecraft attitude control.
Vorticity and the Curl Operator in Fluid Dynamics
In fluid mechanics and aerodynamics, the Curl of a 3D velocity field v(x,y,z) is defined using the formal cross product with the vector differential del operator (∇):
Vorticity quantifies the local microscopic rotation and turbulence of airflow vortices over aircraft wings and turbine blades.
Skew-Symmetric Cross Product Matrix Operator
In robotics kinematics and computer vision, the cross product u × v is expressed as linear matrix multiplication using the Skew-Symmetric Cross Product Matrix [u]×:
This linear matrix transformation enables engineers to formulate continuous rotational equations of motion (Euler's equations of rigid body dynamics) in state-space control models.
Calculating 3D Plane Normal Vectors in 3D Mesh Modeling
In 3D computer graphics, the outward normal vector N of a triangular polygon with vertices A, B, and C is computed via the cross product of two edge vectors: N = (B - A) × (C - A), essential for dynamic shadow and lighting rendering.
Exterior Algebra and 2-Forms in Differential Geometry
While the standard cross product is unique to three dimensions, differential geometers generalize the operation to arbitrary n-dimensional manifolds using the Wedge Product (u ∧ v) in Exterior Algebra. The wedge product constructs an oriented bivector (2-form) representing the infinitesimal oriented surface element, enabling coordinate-free formulations of Maxwell's electromagnetic equations (dF = 0 and d*F = J) across curved four-dimensional spacetime.
Gyroscopic Precession and Angular Momentum Cross Products
In classical spinning top physics and spacecraft gyroscopic attitude control, applying an external torque vector τ induces perpendicular rotational precession governed by the time derivative of angular momentum: τ = dL/dt = Ω × L.
Maxwell's Equations and Electromagnetic Cross Products
In classical electrodynamics, the Poynting vector S = (1 / μ0) (E × B) calculates the directional energy flux density of propagating electromagnetic radiation.