Definition and Notation
Vector Operation
Cross product: binary operation on two vectors in ℝ³ producing a third vector orthogonal to both.
Notation
Denoted by: u × v, where u, v ∈ ℝ³.
Output
Result: vector perpendicular to plane containing u and v.
Geometric Interpretation
Orthogonality
Cross product vector orthogonal to both operands.
Magnitude
Magnitude equals area of parallelogram spanned by u and v.
Plane Relation
Resultant vector normal to plane defined by u and v.
Algebraic Formula
Component-wise Definition
u = (u₁, u₂, u₃), v = (v₁, v₂, v₃)u × v = (u₂v₃ - u₃v₂, u₃v₁ - u₁v₃, u₁v₂ - u₂v₁)Vector Components
Each component computed via determinant of 2×2 submatrix.
Right-Handed System
Coordinates follow right-hand orientation for direction.
Properties of Cross Product
Anticommutativity
u × v = −(v × u)
Distributivity
u × (v + w) = u × v + u × w
Scalar Multiplication
(a u) × v = a (u × v) = u × (a v), a ∈ ℝ
Zero Vector Condition
u × v = 0 if u and v are parallel or one is zero vector.
Right-Hand Rule
Direction Determination
Point index finger along u, middle finger along v; thumb points in direction of u × v.
Orientation
Ensures consistent vector direction in ℝ³.
Left-Hand Rule Contrast
Left-hand rule produces opposite direction; not used in standard math conventions.
Applications
Physics
Torque, angular momentum, magnetic force.
Computer Graphics
Normal vector calculations for lighting and shading.
Engineering
Moment calculations in statics and dynamics.
Mathematics
Area computations, vector projections, and orientation tests.
Computational Methods
Determinant Method
u × v = det| i j k || u₁ u₂ u₃ || v₁ v₂ v₃ |Component-wise Calculation
Direct formula application for faster computation.
Software Implementation
Built-in vector cross functions in MATLAB, Python (NumPy), etc.
Relation to Determinant
Definition via Determinant
Cross product components are cofactors of 3×3 determinant expansion.
Matrix Representation
Cross product as determinant of matrix with standard unit vectors.
Determinant Properties
Linearity and antisymmetry inherited from determinant.
| Matrix | Determinant Value |
|---|---|
| | i j k | | u₁ u₂ u₃ | | v₁ v₂ v₃ | | u × v components |
Orthogonality and Perpendicularity
Dot Product Relation
(u × v) · u = 0 and (u × v) · v = 0 ensuring perpendicularity.
Plane Normal Vector
Cross product defines normal vector to span(u,v).
Verification
Orthogonality test via zero dot product.
Cross Product in Physics
Torque
τ = r × F; torque vector perpendicular to position and force vectors.
Angular Momentum
L = r × p; defines rotational momentum direction and magnitude.
Magnetic Force
F = q (v × B); force on charged particle in magnetic field.
| Physical Quantity | Cross Product Formula |
|---|---|
| Torque (τ) | τ = r × F |
| Angular Momentum (L) | L = r × p |
| Magnetic Force (F) | F = q (v × B) |
Limitations and Constraints
Defined Only in ℝ³
Cross product uniquely defined only in three dimensions.
Non-Associative
(u × v) × w ≠ u × (v × w) generally.
Zero Vector Cases
Parallel vectors yield zero cross product.
Worked Examples
Example 1: Basic Calculation
Given:u = (2, 3, 4)v = (5, 6, 7)Calculate u × v:= (3*7 - 4*6, 4*5 - 2*7, 2*6 - 3*5)= (21 - 24, 20 - 14, 12 - 15)= (-3, 6, -3)Example 2: Orthogonality Check
Verify (u × v) · u = 0:
(-3, 6, -3) · (2, 3, 4) = (-3*2) + (6*3) + (-3*4)= -6 + 18 - 12 = 0Example 3: Area of Parallelogram
Area spanned by u and v equals |u × v|:
|u × v| = sqrt((-3)² + 6² + (-3)²) = sqrt(9 + 36 + 9) = sqrt(54) ≈ 7.348References
- Anton, H., & Rorres, C. Elementary Linear Algebra, Wiley, 11th Edition, 2013, pp. 195-210.
- Strang, G. Introduction to Linear Algebra, Wellesley-Cambridge Press, 5th Edition, 2016, pp. 150-165.
- Axler, S. Linear Algebra Done Right, Springer, 3rd Edition, 2015, pp. 123-134.
- Lay, D.C. Linear Algebra and Its Applications, Pearson, 5th Edition, 2015, pp. 180-195.
- Heath, M.T. Scientific Computing: An Introductory Survey, McGraw-Hill, 2nd Edition, 2002, pp. 75-85.