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Cross Product in LaTeX: Vector Notation and Symbols

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Cross Product in LaTeX: Vector Notation and Symbols

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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