Cross Product in LaTeX: Vector Notation and Symbols
Date Published
Cross Product in LaTeX
Write cross products and vector operations correctly in LaTeX.
Cross Product Symbol
Standard Cross (\times)
$\vec{A} \times \vec{B}$
This is the most common notation for cross product.
Alternative: \wedge (Wedge Product)
$\vec{A} \wedge \vec{B}$
Used in differential geometry and exterior algebra.
Vector Notation
Arrow Over Vector
$\vec{v}$
Bold Vector
$\mathbf{v}$
Bold with Arrow (for emphasis)
$\boldsymbol{\vec{v}}$
Cross Product Formulas
Basic Definition
$\vec{A} \times \vec{B} = |\vec{A}||\vec{B}|\sin\theta \, \hat{n}$
Component Form
$\vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{vmatrix}$
Expanded Form
$\vec{A} \times \vec{B} = (A_y B_z - A_z B_y)\hat{i} - (A_x B_z - A_z B_x)\hat{j} + (A_x B_y - A_y B_x)\hat{k}$
Related Symbols
Dot Product
$\vec{A} \cdot \vec{B}$
Tensor Product
$\vec{A} \otimes \vec{B}$
Magnitude
$|\vec{A}|$ or $\|\vec{A}\|$
Unit Vectors
Cartesian
$\hat{i}, \hat{j}, \hat{k}$ or $\hat{x}, \hat{y}, \hat{z}$
General Unit Vector
$\hat{n}$ or $\hat{e}$
Physics Examples
Torque
$\vec{\tau} = \vec{r} \times \vec{F}$
Magnetic Force
$\vec{F} = q\vec{v} \times \vec{B}$
Angular Momentum
$\vec{L} = \vec{r} \times \vec{p}$
Poynting Vector
$\vec{S} = \frac{1}{\mu_0}\vec{E} \times \vec{B}$
Cross Product Properties
Anti-commutative
$\vec{A} \times \vec{B} = -\vec{B} \times \vec{A}$
Distributive
$\vec{A} \times (\vec{B} + \vec{C}) = \vec{A} \times \vec{B} + \vec{A} \times \vec{C}$
With Scalar
$(c\vec{A}) \times \vec{B} = c(\vec{A} \times \vec{B})$
Parallel Vectors
$\vec{A} \times \vec{A} = \vec{0}$
Triple Products
Scalar Triple Product
$\vec{A} \cdot (\vec{B} \times \vec{C})$
Vector Triple Product
$\vec{A} \times (\vec{B} \times \vec{C})$
BAC-CAB Rule
$\vec{A} \times (\vec{B} \times \vec{C}) = \vec{B}(\vec{A} \cdot \vec{C}) - \vec{C}(\vec{A} \cdot \vec{B})$
Packages for Vectors
physics Package
\usepackage{physics}
Provides: \vb{v} for bold vectors, \cross for cross product
$\vb{A} \cross \vb{B}$
esvect Package
\usepackage{esvect}
Better vector arrows: $\vv{AB}$
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