Cross Product
Vector product in 3D space
About Cross Product
The Cross Product represents vector product in 3D space. This linear algebra formula is fundamental to mathematical analysis and serves as a cornerstone concept that students and professionals encounter throughout their mathematical journey. Its importance extends beyond pure mathematics into applied fields where quantitative analysis is required.
This formula is essential in Linear algebra and Vector calculus. It serves as a building block for more advanced mathematical theory and provides the foundation needed to understand complex mathematical relationships. Whether you're studying mathematics, physics, engineering, or economics, familiarity with this formula enhances your analytical capabilities.
Practical applications of the Cross Product include Physics, Computer graphics, Engineering, among others. Understanding and correctly applying this formula enables problem-solvers to approach challenges more systematically and efficiently. Mastery of this concept not only expands your mathematical knowledge but also improves your overall quantitative reasoning skills.
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LaTeX Code
\vec{a} \times \vec{b} = \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}Formula Information
Difficulty Level
Prerequisites
Discovered
19th century
Discoverer
William Rowan Hamilton
Real-World Applications
Examples
Mathematical Fields
Keywords
Related Topics
Important Notes
Result is perpendicular to both input vectors. Magnitude equals area of parallelogram.
Alternative Names
Common Usage
Formula Variations
Frequently Asked Questions
What is the cross product?
The cross product (a × b) of two 3D vectors produces a third vector perpendicular to both. The magnitude is |a||b|sin(θ), where θ is the angle between them. The direction follows the right-hand rule. The result's magnitude equals the area of the parallelogram formed by a and b.
How do I calculate the cross product?
Use the determinant formula: a × b = |i j k; a₁ a₂ a₃; b₁ b₂ b₃| = (a₂b₃ - a₃b₂, a₃b₁ - a₁b₃, a₁b₂ - a₂b₁). Or use: |a × b| = |a||b|sin(θ) with right-hand rule for direction. The cross product is only defined in 3D (and 7D with a different structure).
What's the right-hand rule?
Point your right hand's fingers in the direction of the first vector (a). Curl them toward the second vector (b). Your thumb points in the direction of a × b. Alternatively: if a points along your index finger and b along your middle finger, your thumb gives a × b. This determines the perpendicular direction.
What are properties of the cross product?
Key properties: 1) Anticommutative: a × b = -(b × a), 2) Not associative: (a × b) × c ≠ a × (b × c), 3) Distributive: a × (b + c) = a × b + a × c, 4) Scalar multiplication: (ka) × b = k(a × b), 5) Orthogonal: (a × b) · a = 0 and (a × b) · b = 0.
What are practical applications?
The cross product is used in: physics (torque τ = r × F, angular momentum L = r × p), computer graphics (normal vectors, lighting calculations), engineering (moment calculations, force analysis), robotics (orientation, rotation), and navigation (coordinate transformations).
What's the relationship between cross product and dot product?
The cross product gives a vector (perpendicular to both inputs), while the dot product gives a scalar (projection). They're related: |a × b|² = |a|²|b|² - (a·b)². Also, (a × b) · c gives the scalar triple product (volume of parallelepiped). Both are fundamental vector operations.
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Quick Details
- Category
- Linear Algebra
- Difficulty
- Intermediate
- Discovered
- 19th century
- Discoverer
- William Rowan Hamilton
- Formula ID
- cross-product