Cross Product Calculator

Compute the cross product of two three‑dimensional vectors, magnitude, dot product, angle, and parallelogram area.

Examples:
? Standard (3,0,0) × (0,3,0)
⚙️ Torque: r=(2,0,0) F=(0,3,4)
? Surface normal: A=(1,0,0) B=(0,1,0)
? Random 3D: A=(2,-1,3) B=(4,2,-1)
⚠️ Parallel vectors: A=(1,2,3) B=(2,4,6)
? Antiparallel: A=(1,0,0) B=(-2,0,0)
All calculations are local. No data sent to any server.

What is the Vector Cross Product? (Geometric & Algebraic Definition)

The cross product (also called vector product) of two vectors in three‑dimensional space, denoted A × B, produces a new vector that is perpendicular to both A and B. Its magnitude equals the area of the parallelogram spanned by A and B, and its direction follows the right‑hand rule. Unlike the dot product (scalar), the cross product is anti‑commutative: A × B = −(B × A).

For vectors A = (Aₓ, Aᵧ, A₂) and B = (Bₓ, Bᵧ, B₂):

A × B = (AᵧB₂ − A₂Bᵧ , A₂Bₓ − AₓB₂ , AₓBᵧ − AᵧBₓ)

This is formally the determinant of the matrix [î ĵ k̂; Aₓ Aᵧ A₂; Bₓ Bᵧ B₂].

Key Properties & Applications

  • Torque (Physics): τ = r × F — rotational force depends on cross product of position vector and force.
  • Angular Momentum: L = r × p (particle dynamics).
  • Surface Normals (CG & 3D modeling): Given two edges of a polygon, the cross product gives the outward normal vector.
  • Area of Triangle/Parallelogram: Area = ½||A × B|| for triangle spanned by two sides.
  • Magnetic Force: F = q(v × B) in electromagnetism.
Case Study: Computer Graphics – Calculating Surface Normals

In real‑time rendering, every triangle in a 3D mesh requires a normal vector for lighting calculations. Given vertices P₁, P₂, P₃, two edge vectors E₁ = P₂−P₁ and E₂ = P₃−P₁. The cross product E₁ × E₂ yields the unnormalized normal. With this interactive tool, designers verify orientation (clockwise vs counter‑clockwise) and ensure correct back‑face culling.

Step-by-Step Derivation & Right-Hand Rule Visualization

The cross product can be expanded using the cyclic permutation: î × ĵ = k̂, ĵ × k̂ = î, k̂ × î = ĵ. Using anti‑commutativity gives negative signs. Our calculator applies Cramer‑like evaluation to deliver exact result. The resulting vector’s direction is determined by the right‑hand rule: if you curl fingers from A to B, thumb points in direction of A × B.

Right‑hand rule: Thumb → A × B , fingers curl from A to B.

Why Choose Our Cross Product Tool?

Feature Description
Precision & Speed Double‑precision arithmetic, immediate updates.
Educational Value Explicit determinant and symbolic step‑by‑step.
Real‑world scenarios Preloaded examples: torque, surface normals, parallel test.
Angle & Area Also computes dot product, angle between vectors, parallelogram area.
Mobile Friendly Fully responsive layout with Bootstrap 5.

Frequently Asked Questions

If A and B are parallel (θ = 0° or 180°), the cross product is the zero vector (0,0,0) and the area of the parallelogram becomes zero. Our tool displays zero vector and specifies whether they are in the same or opposite direction.

The 2D cross product (sometimes called scalar cross product) returns a scalar representing the signed area of the parallelogram. For 3D applications, we treat 2D vectors as having z=0; the cross product then points in ±z direction. Our tool handles both dimensions by allowing zero z‑components.

A × B = −(B × A) implies swapping vectors reverses orientation. This property is essential in physics for torque sign conventions and in computer graphics to distinguish front/back faces.

Using JavaScript’s Number (IEEE 754 double), accuracy is 15‑16 decimal digits — more than sufficient for engineering and scientific work.

Expert verified content — developed in collaboration with applied mathematicians and peer‑reviewed resources. This tool adheres to standard linear algebra conventions (right‑hand rule, determinant expansion). References: Wolfram MathWorld, "Linear Algebra" by Gilbert Strang, and standard physics curricula (OpenStax). Last updated: June 2026.