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Cross Product Calculator

Last updated: November 11, 2025
Reviewed by: LumoCalculator Team

Calculate the cross product (vector product) of two 3D vectors. Find the perpendicular vector, magnitude, direction, and visualize step-by-step calculations for vector analysis in mathematics, physics, and engineering.

Calculate Cross Product

Vector A = a₁i + a₂j + a₃k
Vector B = b₁i + b₂j + b₃k

Cross Product Result

A × B =-3i + 6j - 3k(-3, 6, -3)
Magnitude |A × B|
7.3485
Parallelogram Area
7.3485
Unit Vector (û)
-0.4082i + 0.8165j - 0.4082k
✓ Orthogonality Verified

Result vector is perpendicular to both input vectors.

Calculation Steps

Determinant Form:
| i    j    k  |
| 1  2  3 |
| 4  5  6 |
i: (2)(6) - (3)(5) = 12 - 15 = -3
j: -[(1)(6) - (3)(4)] = -[6 - 12] = 6
k: (1)(5) - (2)(4) = 5 - 8 = -3
Result: -3i + 6j -3k

Related Calculators

What is Cross Product?

The cross product (also called vector product) is a binary operation on two vectors in three-dimensional space. Unlike the dot product which yields a scalar, the cross product produces a new vector that is perpendicular to both original vectors.

Key Characteristics

  • ✓Direction: Perpendicular to both input vectors (right-hand rule)
  • ✓Magnitude: |A × B| = |A||B|sin(θ), area of parallelogram
  • ✓Not commutative: A × B = -(B × A)
  • ✓Zero when parallel: A × B = 0 if vectors are parallel

How to Calculate Cross Product

Cross Product Formula

A × B = (a₂b₃ - a₃b₂)i - (a₁b₃ - a₃b₁)j + (a₁b₂ - a₂b₁)k
Or using determinant:
A × B = | i    j    k  |
        | a₁   a₂   a₃ |
        | b₁   b₂   b₃ |

Calculation Steps:

  1. 1
    Identify vector components
    Write A = (a₁, a₂, a₃) and B = (b₁, b₂, b₃)
  2. 2
    Calculate i-component
    i: a₂b₃ - a₃b₂ (cross-multiply middle and last)
  3. 3
    Calculate j-component
    j: -(a₁b₃ - a₃b₁) (negative, first and last)
  4. 4
    Calculate k-component
    k: a₁b₂ - a₂b₁ (cross-multiply first and middle)

Important Considerations

⚠️ Mathematical Note

Cross product is only defined for 3D vectors. For higher dimensions, use wedge product. Always verify orthogonality of result.

🔄 Not Commutative

Order matters in cross product

  • • A × B = -(B × A)
  • • Reversing order flips direction
  • • Magnitude stays the same
📐 Right-Hand Rule

Determines result direction

  • • Point fingers along first vector
  • • Curl toward second vector
  • • Thumb shows result direction
⊥ Orthogonality

Result perpendicular to inputs

  • • (A × B) · A = 0
  • • (A × B) · B = 0
  • • Verify with dot product
⚡ Parallel Vectors

Special case with zero result

  • • A × A = 0 (same vector)
  • • Parallel vectors → zero
  • • No unique perpendicular

Properties of Cross Product

Anti-Commutativity

A × B = -(B × A)

Swapping vectors changes sign but not magnitude

Distributive Property

A × (B + C) = A × B + A × C

Cross product distributes over vector addition

Scalar Multiplication

(kA) × B = k(A × B) = A × (kB)

Scalars can be factored out

Lagrange Identity

|A × B|² = |A|²|B|² - (A · B)²

Relates cross product to dot product

Example Cases

Case 1: Standard Unit Vectors

Input:
A = i = (1, 0, 0)
B = j = (0, 1, 0)
Result:
A × B = k = (0, 0, 1)
|A × B| = 1

Use Case: Demonstrates right-hand rule with coordinate axes. i × j = k forms orthonormal basis.

Case 2: Physics - Torque Calculation

Input:
r = (2, 1, 0) m (position)
F = (0, 0, 10) N (force)
Result:
τ = r × F = (10, -20, 0) N·m
|τ| = 22.36 N·m

Use Case: Finding torque about origin. Direction shows axis of rotation per right-hand rule.

Case 3: Geometry - Surface Normal

Input:
A = (1, 2, 3) (edge 1)
B = (4, 5, 6) (edge 2)
Result:
A × B = (-3, 6, -3)
Area = 7.35 square units

Use Case: Finding normal vector to plane defined by two edges. Magnitude gives parallelogram area.

Applications of Cross Product

⚙️

Physics & Engineering

  • • Torque: τ = r × F
  • • Angular momentum: L = r × p
  • • Magnetic force: F = q(v × B)
  • • Angular velocity: v = ω × r
📐

Geometry

  • • Surface normals: 3D graphics
  • • Area calculation: Parallelograms
  • • Plane equations: Normal vectors
  • • Volume: Triple scalar product
🎮

Computer Graphics

  • • Lighting: Normal vector calculation
  • • Backface culling: View direction
  • • Collision detection: Plane tests
  • • Camera systems: View transformations
🚀

Robotics & Navigation

  • • Rotation axes: Orientation control
  • • Force couples: Mechanical systems
  • • Gimbal lock: Avoidance strategies
  • • Coordinate transforms: Frame rotations

Frequently Asked Questions

What is the cross product of two vectors?
The cross product (or vector product) of two 3D vectors A and B produces a new vector that is perpendicular to both input vectors. The magnitude of the cross product equals the area of the parallelogram formed by the two vectors. It's denoted as A × B and follows the right-hand rule for direction.
How do I calculate cross product step by step?
To calculate A × B: (1) Write vectors in component form A = (a₁, a₂, a₃) and B = (b₁, b₂, b₃). (2) Set up a 3×3 determinant with i, j, k in the first row. (3) Calculate: i-component = a₂b₃ - a₃b₂, j-component = -(a₁b₃ - a₃b₁), k-component = a₁b₂ - a₂b₁. (4) Combine: (i-comp)i + (j-comp)j + (k-comp)k.
What does it mean when cross product is zero?
When the cross product of two vectors is zero (A × B = 0), it means the vectors are parallel (pointing in the same or opposite directions) or at least one vector is the zero vector. This occurs because parallel vectors don't span a parallelogram with non-zero area, resulting in zero magnitude.
What is the right-hand rule for cross product?
The right-hand rule determines the direction of A × B: Point your right hand's fingers along vector A, curl them toward vector B, and your thumb points in the direction of A × B. Note that B × A points in the opposite direction, demonstrating that cross product is not commutative (A × B ≠ B × A).
How is cross product used in physics?
Cross product is fundamental in physics for calculating: (1) Torque: τ = r × F (position vector × force), (2) Angular momentum: L = r × p (position × momentum), (3) Magnetic force: F = q(v × B) (charge × velocity × magnetic field), and (4) Normal vectors to surfaces in 3D space.
What's the difference between dot product and cross product?
Dot product (A · B) produces a scalar representing projection and is commutative. Cross product (A × B) produces a vector perpendicular to both inputs and is not commutative. Dot product measures parallel components, while cross product measures perpendicular span. Both are zero when vectors are parallel, but dot product is also zero when vectors are perpendicular.