Cross Product

Prerequisites: Vector Calculus. Full path (3 pages)

• Intuitive
• Concrete
• Abstract
• Why is it interesting?

Intuitive

Two arrows that start at the same point span a parallelogram. The cross product turns this parallelogram into a single new arrow. The new arrow stands perpendicular to the parallelogram, and its length equals the parallelogram's area. Which of the two perpendicular directions it takes is fixed by the right-hand rule: point the fingers of your right hand along the first arrow, curl them towards the second, and your thumb gives the direction of the result.

So the cross product is large when the two arrows are long and at right angles to each other, and zero when they point along the same line, because then the parallelogram is flat. Swapping the order of the two arrows flips the result.

This is what physics needs whenever something turns. Pushing a door with a force at some distance from the hinges produces a torque, and the torque arrow points along the axis about which the door rotates. Its size grows with both the force and the distance, and it is largest when you push at right angles to the door. Angular momentum and the magnetic force on a moving charge are built the same way.

Concrete

• Dot product, the interactions between similar dimensions (x*x y*y, z*z)
• Cross product, the interactions between different dimensions (x*y,y*z, z*x, etc.)

The dot product (vec(a) · vec(b)) measures similarity because it only accumulates interactions in matching dimensions. It’s a simple calculation with 3 components.

The cross product (written vec(a) times vec(b)) has to measure a half-dozen “cross interactions”. The calculation looks complex but the concept is simple: accumulate 6 individual differences for the total.

Vector Calculus: Understanding the Dot Product by Kalid Azad

Recommended Resources:

• Vector Calculus: Understanding the Dot Product by Kalid Azad
• A Student's Guide to Vectors and Tensors by Daniel A. Fleisch
• No-Nonsense Electrodynamics by Jakob Schwichtenberg, section "The cross product"

Abstract

In components, with the Levi-Civita symbol $\epsilon_{ijk}$ ($\epsilon_{123} = 1$, totally antisymmetric) and summation over repeated indices,

$$(\vec a \times \vec b)_i = \epsilon_{ijk}\, a_j b_k .$$

Its definition uses both the Euclidean metric and an orientation of $\mathbb{R}^3$. In the language of differential forms, the natural antisymmetric product of two vectors is the exterior product $a \wedge b \in \Lambda^2\mathbb{R}^3$, which encodes the oriented plane and area of the parallelogram. The cross product is its Hodge dual, $\vec a \times \vec b = \star(a \wedge b)$, which is possible only because $\Lambda^2\mathbb{R}^3$ and $\mathbb{R}^3$ both have dimension 3.

Behaviour under reflections

For any $R \in O(3)$,

$$(R\vec a) \times (R\vec b) = (\det R)\, R(\vec a \times \vec b) .$$

For rotations, $\det R = 1$, and the cross product is covariant. Under parity $\vec x \to -\vec x$, both factors change sign but the product does not. Quantities like the angular momentum $\vec L = \vec x \times \vec p$ and the magnetic field are therefore pseudovectors (axial vectors).

Lie algebra structure

The cross product satisfies $\vec a \times \vec b = -\vec b \times \vec a$ and the Jacobi identity

$$\vec a \times (\vec b \times \vec c) + \vec b \times (\vec c \times \vec a) + \vec c \times (\vec a \times \vec b) = 0 ,$$

so $(\mathbb{R}^3, \times)$ is a Lie algebra. It is isomorphic to $\mathfrak{so}(3)$, the Lie algebra of the rotation group, via $\vec a \mapsto \hat a$ with $(\hat a)_{ij} = -\epsilon_{ijk} a_k$. Then $\hat a\, \vec b = \vec a \times \vec b$ and $[\hat a, \hat b] = \widehat{\vec a \times \vec b}$. The infinitesimal rotation about the axis $\vec\omega$ acts as $\delta\vec x = \vec\omega \times \vec x\, \delta t$, which is why the velocity $\vec v = \vec\omega \times \vec x$ of a point on a rotating body is a cross product. See also SO(3).

Other dimensions

A bilinear product $\mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^n$ that is orthogonal to both factors and satisfies $|\vec a \times \vec b|^2 = |\vec a|^2|\vec b|^2 - (\vec a\cdot\vec b)^2$ exists, for $n \ge 2$, only for $n = 3$ and $n = 7$. The seven-dimensional one comes from the imaginary octonions and does not satisfy the Jacobi identity. In other dimensions one works with the exterior product directly.

Why is it interesting?

The cross product is a tool that we can use to combine two vectors and get another vector out. This vector describes the plane spanned by the two vectors. The magnitude of the resulting vector is the area of the parallelogram spanned by them, and the direction denotes the orientation of the plane, because it is perpendicular to it.

This is an extremely useful concept and used in almost any physical theory, like for example, electrodynamics. Moreover, many other important tools, like the Curl are defined with the help of the cross product.

Path to this page

1. Calculus
2. Linear Algebra
3. Vector Calculus