# Cartesian Product: \mathbb{R}^3

• quasar987
In summary, the elements of the cartesian product of two sets, \mathbb{R}\times \mathbb{R}^2, are of the form (a, (b, c)), while the elements of the cartesian product of three sets, \mathbb{R}^2\times \mathbb{R}, are of the form ((a, b), c). However, they are considered isomorphic and are often referred to as simply \mathbb{R}^3 for convenience. This concept can also be expressed using the language of category theory.
quasar987
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Is it true that

$$\mathbb{R}\times \mathbb{R}^2 = \mathbb{R}^2 \times \mathbb{R} = \mathbb{R}^3$$

?

Yes. (with a but)

Wouldn't the elements of $$\mathbb{R}\times\mathbb{R}^2$$ all be of the form (a, (b, c)) whereas the elements of $$\mathbb{R}^2\times\mathbb{R}$$ all be of the form ((a, b), c)?

(Assuming you meant the 2 to have precedence over the x)

That's the essense of the "but".

But how important is this "but"?

It's so unimportant, I didn't feel the need to elaborate on it in my response.

In the strictest sense, they aren't equal, but I don't think I've ever really seen anyone distinguish between them.

Strictly speaking they are isomorphic not equal, however the isomoprphism is fairly 'canonical' and in general there are canonical isomorphisms between Ax(BxC) and (AxB)xC, and we will by commonly accepted abuse of notation refer to it as AxBxC.

This is one of the "modern" ways of saying it in the lagauge of category theory.

## 1. What is a Cartesian Product?

A Cartesian Product is a mathematical operation that combines two sets to create a new set. It is denoted by the symbol x and is also known as a cross product. In terms of geometry, it represents the combination of two coordinate systems to create a new coordinate system.

## 2. What is the significance of "R^3" in the Cartesian Product?

The symbol "R^3" represents the set of all ordered triples of real numbers, which is also known as the three-dimensional Cartesian coordinate system. It is commonly used to represent points in three-dimensional space and is an essential concept in geometry, physics, and engineering.

## 3. How is the Cartesian Product related to vectors?

The Cartesian Product is closely related to vectors as it can be used to represent and manipulate vectors in three-dimensional space. The cross product of two vectors results in a new vector that is perpendicular to both input vectors and has a magnitude equal to the product of their magnitudes.

## 4. What are some real-world applications of the Cartesian Product in science?

The Cartesian Product has many practical applications in science, including physics, engineering, and computer graphics. It is used to represent the motion of objects in three-dimensional space, calculate forces and velocities, and create three-dimensional models of objects and structures.

## 5. Can the Cartesian Product be extended to more than three dimensions?

Yes, the Cartesian Product can be extended to any number of dimensions. For example, the Cartesian Product of four sets would result in a four-dimensional space, and so on. This concept is used in higher mathematics and has applications in various fields of science and engineering.

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