Cross Product Calculator
Calculate Vector Cross Product
Compute cross product of two 3D vectors. Find perpendicular vector, magnitude, area of parallelogram, and angle between vectors.
Cross Product Result
(-3, 6, -3)
Magnitude of A × B
Angle Between Vectors
Step-by-Step Calculation:
Vector Visualization:
Cross Product Properties:
The cross product of two vectors produces a vector perpendicular to both, with magnitude equal to the area of the parallelogram they span.
What is Cross Product?
The cross product (also called vector product) is a binary operation on two vectors in three-dimensional space. It results in a vector that is perpendicular to both original vectors, with magnitude equal to the area of the parallelogram that the vectors span. The cross product is widely used in physics, engineering, and computer graphics.
Cross Product Formulas
Component Formula
Direct calculation
Using vector components
Determinant Formula
Matrix determinant
Formal definition
Magnitude Formula
Geometric interpretation
Area of parallelogram
Right-Hand Rule
Direction determination
Coordinate system
Mathematical Definitions
1. Component Formula
2. Determinant Method
3. Geometric Properties
Area = |A × B|
Direction: Right-hand rule
Vector Operations Comparison
| Operation | Notation | Result Type | Properties |
|---|---|---|---|
| Cross Product | A × B | Vector | Perpendicular to both, magnitude = area |
| Dot Product | A · B | Scalar | Measures projection, A·B = |A||B|cosθ |
| Scalar Multiplication | kA | Vector | Changes magnitude, same direction |
| Vector Addition | A + B | Vector | Component-wise addition |
Real-World Applications
Physics & Engineering
- Torque calculation: τ = r × F (position vector cross force)
- Angular momentum: L = r × p (position cross linear momentum)
- Magnetic force: F = q(v × B) (charge velocity cross magnetic field)
- Electromagnetism: Maxwell's equations and electromagnetic waves
Computer Graphics & 3D Modeling
- Surface normals: Calculating normal vectors for lighting
- Camera orientation: Determining view direction and up vectors
- Collision detection: Finding penetration depth and direction
- Rotation axes: Determining axis of rotation between vectors
Mechanics & Robotics
- Robot arm dynamics: Calculating forces and moments
- Flight dynamics: Aircraft orientation and control surfaces
- Vehicle dynamics: Steering forces and stability analysis
- Structural analysis: Stress and strain calculations
Everyday Examples
- Opening a door: The force applied creates torque (cross product)
- Using a wrench: Force perpendicular to wrench creates maximum torque
- Bicycle turning: Angular momentum and steering forces
- Electric motors: Magnetic fields interacting with current
Cross Product Properties
| Property | Formula | Description | Example |
|---|---|---|---|
| Anti-commutative | A × B = - (B × A) | Order matters, direction reverses | i × j = k, j × i = -k |
| Distributive | A × (B + C) = A×B + A×C | Distributes over addition | Valid for all vectors |
| Scalar Multiplication | (kA) × B = k(A × B) | Scalars can be factored out | 2(i×j) = 2k |
| Parallel Vectors | A × B = 0 if A∥B | Zero cross product | (1,2,3) × (2,4,6) = (0,0,0) |
Step-by-Step Calculation Examples
Example 1: Basic Calculation
Given: A = (1, 2, 3), B = (4, 5, 6)
- Calculate x-component: a₂b₃ - a₃b₂ = (2×6) - (3×5) = 12 - 15 = -3
- Calculate y-component: a₃b₁ - a₁b₃ = (3×4) - (1×6) = 12 - 6 = 6
- Calculate z-component: a₁b₂ - a₂b₁ = (1×5) - (2×4) = 5 - 8 = -3
- Result: A × B = (-3, 6, -3)
- Verify: |A × B| = √(9 + 36 + 9) = √54 ≈ 7.35
Example 2: Perpendicular Unit Vectors
Given: i = (1, 0, 0), j = (0, 1, 0)
- Calculate x-component: 0×0 - 0×1 = 0 - 0 = 0
- Calculate y-component: 0×0 - 1×0 = 0 - 0 = 0
- Calculate z-component: 1×1 - 0×0 = 1 - 0 = 1
- Result: i × j = (0, 0, 1) = k
- Verify right-hand rule: i→j→k forms right-handed system
Example 3: Parallel Vectors
Given: A = (2, 4, 6), B = (1, 2, 3) (B = ½A)
- Calculate x-component: 4×3 - 6×2 = 12 - 12 = 0
- Calculate y-component: 6×1 - 2×3 = 6 - 6 = 0
- Calculate z-component: 2×2 - 4×1 = 4 - 4 = 0
- Result: A × B = (0, 0, 0)
- Interpretation: Parallel vectors have zero cross product
Right-Hand Rule Visualization
Related Calculators
Frequently Asked Questions (FAQs)
Q: What's the difference between cross product and dot product?
A: Cross product gives a vector perpendicular to both inputs (A × B = vector). Dot product gives a scalar measuring projection (A · B = scalar). Cross product magnitude equals area of parallelogram, dot product relates to cosine of angle.
Q: Why is cross product only defined in 3D?
A: In 3D space, there's exactly one direction perpendicular to two non-parallel vectors. In 2D, no unique perpendicular direction exists. In higher dimensions, multiple perpendicular directions exist, so cross product as defined in 3D doesn't generalize directly.
Q: How do I determine the direction of the cross product?
A: Use the right-hand rule: Point fingers in direction of first vector, curl toward second vector, thumb points in direction of cross product. Alternatively, use the determinant method with basis vectors i, j, k.
Q: What does a zero cross product mean?
A: Zero cross product (0,0,0) means either: (1) One or both vectors are zero vectors, or (2) The vectors are parallel or anti-parallel (θ = 0° or 180°).
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