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Cross Product: Introduction

Recall our discussion on multiplying vectors. When we tried to invent a way to multiply vectors, we said that there are two ways we can multiply vectors: one that results in a scalar and one that results in a vector. The "scalar product" is the dot product, and the "vector product" is our current topic: the cross product. We will try to think of certain conditions that we need from a vector product, and see how that leads us to the cross product.

Table of Contents​

Introduction​

The cross product is a vector product, which means that when we multiply two vectors, we get another vector. Let's consider what we need from a vector product:

  1. The product should, obviously, be a vector.
  2. All pairs of vectors should have a product; it should be defined for all vectors.
  3. The product should output one vector; there should be no ambiguity in the result.
  4. The product should follow the usual rules of multiplication, such as:

The cross product satisfies all these conditions and, in addition, has some interesting geometric properties. Just like when discussing the dot product, we will first explore the geometry of the cross product. Unlike the dot product, the cross product gives us a vector, so we will have to consider both the magnitude and direction of the cross product.

In the following two sections, we will discuss the magnitude and direction of the cross product of and .

Magnitude​

Recall that we want the cross product to obey the "usual rules of multiplication". For instance, if we double the magnitude of one of the vectors, the magnitude of the cross product should also double.

Let's take a geometric approach; consider the parallelogram spanned by and . We can define the area of this parallelogram as the magnitude of the cross product of and .

If we double the magnitude of , the parallelogram will also double in area. This means that the magnitude of the cross product should also double.

Given that the angle between and is , the magnitude of the cross product can be calculated with some trigonometry:

Direction​

This is where the cross product gets interesting. Our condition for the cross product was that it should output one vector. Additionally, there should be a one-to-one correspondence between the directions of the input vectors and the output vector. This means that changing the directions of and should also change the direction of the cross product.

One clever way to make the cross product depend on the directions of and is to make it perpendicular to both vectors. Imagine and as two vectors that define a plane. Then, the cross product should be perpendicular to this plane; it should "stick out" of the plane:

However, there are two directions perpendicular to a plane - one pointing "up" and one pointing "down". We need a way to choose one of these directions without ambiguity. One way to do this is to use the right-hand rule. This rule is a convention, and it is used to ensure that the cross product is well-defined:

The direction of the cross product is given by the right-hand rule: if you curl the fingers of your right hand from to , your thumb points in the direction of .

Note that this has an interesting consequence: the cross product is not commutative. If you swap the order of the vectors, the direction of the cross product changes. Instead, the cross product is anticommutative:

Also, this means that the cross product is only defined in three dimensions. In lower dimensions, like 2D, there is no unique direction perpendicular to a plane, so the cross product is not defined. In higher dimensions, there are multiple directions perpendicular to a plane, so the cross product is not unique. 3D is the "sweet spot" where the cross product is well-defined.

Summary and Next Steps​

In this page, we introduced the cross product as a way to multiply vectors. We discussed the conditions that a vector product should satisfy and saw how the cross product meets these conditions. We also explored the magnitude and direction of the cross product and discussed the right-hand rule.

Here are the key points to remember:

  • The cross product is a vector product that takes two vectors and outputs a vector.

  • The magnitude of the cross product is the area of the parallelogram spanned by the two vectors:

  • The direction of the cross product is given by the right-hand rule.

If all of this screams "determinant" to you, you're on the right track. In the next section, we will explore the cross product in terms of determinants and matrices, and see how this leads us to an explicit formula for the cross product.

Appendix: The Geometric Product​

We discussed the cross product as a vector product, but there is another way to multiply vectors that is even more general. Imagine if we can combine the dot product and the cross product into a single operation. This operation is called the geometric product and is simply denoted by .

The geometric product has many interesting properties, and it is the basis of a mathematical system called geometric algebra, or Clifford algebra. In Clifford algebra, we can represent scalars, vectors, and even higher-dimensional objects like planes and volumes using a single mathematical object called a multivector. This system is very powerful and has applications in physics, computer graphics, and robotics.

The geometric product is a very deep topic, and we will not delve into it in this note. We will just discuss two important things about the geometric product:

  1. In Clifford algebra, cross products are replaced by the wedge product. Since Clifford algebra allows us to represent planes, it makes sense for the product to just be a plane (the parallelogram spanned by the vectors). The wedge product is denoted by .

  2. The geometric product is the sum of the dot product and the wedge product: