LaTeX Quick Guides

Cross Product in LaTeX: Complete Guide

November 5, 202512 min read

Introduction

The cross product (also known as the vector product) is a fundamental operation in linear algebra and vector calculus. It takes two vectors in three-dimensional space and produces a third vector that is perpendicular to both input vectors. This operation is essential in physics, engineering, and computer graphics.

In LaTeX, typesetting the cross product requires proper vector notation, the cross product symbol, and sometimes matrix determinant notation. This complete guide covers everything you need to write the cross product correctly in LaTeX, from basic notation to advanced applications. This comprehensive guide provides all the essential information for mastering cross product notation in LaTeX.

Basic Cross Product Notation

The cross product of two vectors \mathbf{a} and \mathbf{b} is denoted as:

a \times b

The result is a vector that is perpendicular to both \mathbf{a} and \mathbf{b}, following the right-hand rule.

LaTeX Code for Basic Form

\mathbf{a} \times \mathbf{b}

Determinant Form of the Cross Product

The cross product can be expressed using a determinant notation, which is particularly useful for calculations:

a \times b = i a_{1} b_{1} j a_{2} b_{2} k a_{3} b_{3}

Where \mathbf{i}, \mathbf{j}, and \mathbf{k} are the unit vectors along the x, y, and z axes respectively.

Expanded Form

Expanding the determinant gives us the component form:

a \times b = (a_{2} b_{3} - a_{3} b_{2}) i + (a_{3} b_{1} - a_{1} b_{3}) j + (a_{1} b_{2} - a_{2} b_{1}) k

\mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}

Component Form

When working with explicit vector components, the cross product can be written as:

a \times b = a_{2} b_{3} - a_{3} b_{2} a_{3} b_{1} - a_{1} b_{3} a_{1} b_{2} - a_{2} b_{1}

Or using angle bracket notation:

a \times b = ⟨ a_{2} b_{3} - a_{3} b_{2}, a_{3} b_{1} - a_{1} b_{3}, a_{1} b_{2} - a_{2} b_{1} ⟩

LaTeX Code for Component Form

\begin{pmatrix} a_2 b_3 - a_3 b_2 \\ a_3 b_1 - a_1 b_3 \\ a_1 b_2 - a_2 b_1 \end{pmatrix}

\langle a_2 b_3 - a_3 b_2, a_3 b_1 - a_1 b_3, a_1 b_2 - a_2 b_1 \rangle

Magnitude and Geometric Properties

Magnitude of Cross Product

The magnitude of the cross product is given by:

∣ a \times b ∣ = ∣ a ∣∣ b ∣ sin θ

Where \theta is the angle between vectors \mathbf{a} and \mathbf{b}.

Area of Parallelogram

The magnitude of the cross product equals the area of the parallelogram formed by the two vectors:

Area = ∣ a \times b ∣

Orthogonality

The cross product is orthogonal to both input vectors:

(a \times b) \cdot a = 0 and (a \times b) \cdot b = 0

Important Properties

Anticommutativity

The cross product is anticommutative:

a \times b = - b \times a

Distributivity

The cross product distributes over vector addition:

a \times (b + c) = a \times b + a \times c

Scalar Multiplication

Scalar multiplication is associative:

(c a) \times b = c (a \times b) = a \times (c b)

Triple Product Identity

The scalar triple product identity:

a \cdot (b \times c) = b \cdot (c \times a) = c \cdot (a \times b)

Practical Examples

Example 1: Standard Unit Vectors

Cross products of standard unit vectors:

i \times j = k, j \times k = i, k \times i = j

Example 2: Explicit Vector Calculation

Calculate \mathbf{a} \times \mathbf{b} where \mathbf{a} = \langle 1, 2, 3 \rangle and \mathbf{b} = \langle 4, 5, 6 \rangle:

a \times b = i 14 j 25 k 36 = (2 \cdot 6 - 3 \cdot 5) i - (1 \cdot 6 - 3 \cdot 4) j + (1 \cdot 5 - 2 \cdot 4) k

= - 3 i + 6 j - 3 k = ⟨ - 3, 6, - 3 ⟩

Example 3: Parallel Vectors

If two vectors are parallel, their cross product is the zero vector:

a ∥ b \Rightarrow a \times b = 0

Applications in Physics and Engineering

Torque

In physics, torque is calculated using the cross product:

τ = r \times F

Where \mathbf{r} is the position vector and \mathbf{F} is the force vector.

Angular Momentum

Angular momentum is defined as:

L = r \times p

Where \mathbf{p} is the linear momentum.

Magnetic Force

The force on a charged particle in a magnetic field:

F = q (v \times B)

Where q is charge, \mathbf{v} is velocity, and \mathbf{B} is the magnetic field.

LaTeX Typesetting Tips

Vector Notation

Use \mathbf{} for bold vectors or \vec{} for arrows:

\mathbf{a} \times \mathbf{b}

\vec{a} \times \vec{b}

Determinant Notation

Use \begin{vmatrix} for determinant notation:

\begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}

Cross Product Symbol

The cross product symbol \times is the standard LaTeX command. For alternative notation, you can use:

\times - Standard cross product symbol

\wedge - Alternative wedge notation (sometimes used)

Conclusion

The cross product is a fundamental operation in vector algebra, and knowing how to typeset it correctly in LaTeX is essential for mathematical writing in linear algebra, physics, and engineering. Whether you're using the determinant form, component form, or basic notation, proper LaTeX formatting ensures clarity and professionalism.

Remember to use appropriate vector notation (\mathbf{} or \vec{}), choose the right matrix environment for determinants, and maintain consistent formatting throughout your document.