Cross Product Calculator
Calculate the cross product of two 3D vectors with step-by-step solutions
Calculate Cross Product
Vector A = (ax, ay, az)
Vector B = (bx, by, bz)
Cross Product Results
Step-by-Step Calculation
Formula: A × B = (a₂b₃ - a₃b₂, a₃b₁ - a₁b₃, a₁b₂ - a₂b₁)
1. x-component: (3)(4) - (7)(2) = 12.000 - 14.000 = -2.000
2. y-component: (7)(1) - (2)(4) = 7.000 - 8.000 = -1.000
3. z-component: (2)(2) - (3)(1) = 4.000 - 3.000 = 1.000
4. Magnitude: √(-2.000² + -1.000² + 1.000²) = 2.449490
Properties & Verification
• Cross product is perpendicular to both original vectors
• A·(A×B) = 0.000000 ≈ 0 ✓
• B·(A×B) = 0.000000 ≈ 0 ✓
• |A×B| = |A| × |B| × sin(θ) = 7.874 × 4.583 × sin(3.9°) = 2.449490
🤚 Right-Hand Rule
Point your index finger in the direction of vector A, middle finger in the direction of vector B. Your thumb will point in the direction of A × B. This helps visualize the cross product direction!
Example: 3D Vector Cross Product
Given Vectors
Vector A: (2, 3, 7)
Vector B: (1, 2, 4)
Goal: Find A × B (cross product)
Step-by-Step Solution
Using formula: A × B = (a₂b₃ - a₃b₂, a₃b₁ - a₁b₃, a₁b₂ - a₂b₁)
1. x-component: (3)(4) - (7)(2) = 12 - 14 = -2
2. y-component: (7)(1) - (2)(4) = 7 - 8 = -1
3. z-component: (2)(2) - (3)(1) = 4 - 3 = 1
4. Result: A × B = (-2, -1, 1)
5. Magnitude: |A × B| = √((-2)² + (-1)² + 1²) = √6 ≈ 2.449
Cross Product Properties
Anti-Commutative
A × B = -(B × A)
Distributive
A × (B + C) = A × B + A × C
Scalar Multiplication
k(A × B) = (kA) × B = A × (kB)
Parallel Vectors
A × A = 0 (zero vector)
Key Formulas
Component Form
A × B = (a₂b₃-a₃b₂, a₃b₁-a₁b₃, a₁b₂-a₂b₁)
Magnitude
|A × B| = |A| × |B| × sin(θ)
Geometric Meaning
Area of parallelogram formed by A and B
Direction
Perpendicular to both A and B (right-hand rule)
Understanding Cross Product
What is Cross Product?
The cross product (vector product) is a binary operation on two vectors in 3D space that produces a third vector perpendicular to both original vectors. The magnitude equals the area of the parallelogram formed by the two vectors.
Mathematical Definition
For vectors A = (a₁, a₂, a₃) and B = (b₁, b₂, b₃), the cross product is:
Geometric Interpretation
- •Direction: Perpendicular to both input vectors (right-hand rule)
- •Magnitude: Area of parallelogram formed by the vectors
- •Zero result: When vectors are parallel or antiparallel
Applications
Physics
Torque, angular momentum, magnetic forces
Engineering
Rotational mechanics, computer graphics
Geometry
Surface normals, area calculations
Computer Graphics
3D transformations, lighting calculations
🔍 Key Insight
The cross product only exists in 3D (and 7D) space. In 2D, use the scalar cross product which gives the z-component: a₁b₂ - a₂b₁
⚠️ Important Note
Cross product is not commutative: A × B ≠ B × A. Instead, A × B = -(B × A). Order matters for direction!