Descartes’ analytical method, which allows the Euclidean plane to be represented as the Cartesian product $ \mathbb{R}^2 $ through the theory of real numbers, also makes it possible to represent Euclidean space as the Cartesian product $ \mathbb{R}^3 = \mathbb{R}^2 \times \mathbb{R} $. This includes the alternative perspectives of points or vectors, and the natural dot product, which, by representing elements of the space with coordinates, enables the calculation of the distance between two points as the norm of a vector.

1. The Set $\mathbb{R}^3$ as a Representation of Euclidean Space

1.1. The Analytical Representation $\mathbb{R}^2$ of the Euclidean Plane

Descartes’ analytical approach allows the Euclidean plane to be represented as the Cartesian product $\mathbb{R}^2$. The elements of the plane are thus pairs $(x, y)$ of real numbers, considered either as points or as vectors. When considered as vectors, they can be uniquely represented in the form $(x, y) = x \cdot (1, 0) + y \cdot (0, 1)$: we say that the vectors $\vec{i} = (1, 0)$ and $\vec{j} = (0, 1)$ form a basis (called the canonical basis) of the vector plane $\mathbb{R}^2$.

The order in which these two vectors are considered is important: the basis $(\vec{i}, \vec{j})$ determines one orientation (direct) of the Euclidean plane, while the basis $(\vec{j}, \vec{i})$ determines the opposite orientation (indirect) of the plane. The direct orientation is the one where oriented angles are considered to have a positive measure in the counterclockwise direction.

1.2. The Analytical Approach to Representing Space

Using the same approach, the three-dimensional Euclidean space can be represented as a Cartesian product, namely $\mathbb{R}^3 = \mathbb{R}^2 \times \mathbb{R}$. The elements of $\mathbb{R}^3$ are triplets $((x, y), z)$ of real numbers, that is, pairs formed by a pair $(x, y) \in \mathbb{R}^2$ and an element $z \in \mathbb{R}$. To simplify this description, we write such a triplet as $(x, y, z)$, and again consider the elements of $\mathbb{R}^3$ either as points or as vectors.

It is important to understand that points and vectors here are simply two different ways of considering the same objects (geometric for points, algebraic for vectors). As in the Euclidean plane, a vector $(x, y, z) \in \mathbb{R}^3$ can be uniquely represented as $(x, y, z) = x \cdot (1, 0, 0) + y \cdot (0, 1, 0) + z \cdot (0, 0, 1)$, and the vectors $\vec{i} = (1, 0, 0)$, $\vec{j} = (0, 1, 0)$, and $\vec{k} = (0, 0, 1)$ form the canonical basis of the vector space $\mathbb{R}^3$.

Here too, the order $(\vec{i}, \vec{j}, \vec{k})$ in which these vectors are considered determines an orientation of the space.

2. The Natural Dot or Scalar Product in Euclidean Space $\mathbb{R}^3$

In the Euclidean plane, we mentioned the existence of a dot product of two vectors, which is a real number providing geometric information. This natural dot product is closely associated with Euclidean distance, orthogonality, and the Pythagorean theorem, and it is remarkable to find a complete description of these concepts derived from the simple Cartesian product $\mathbb{R}^2$. It is possible to “extend” this operation to a natural dot product of two vectors in the space $\mathbb{R}^3$, which is also associated with orthogonality and distances.

2.1. Analytical Definition, Euclidean Norm, and Distance

If $\vec{u} = (x, y, z)$ and $\vec{v} = (a, b, c)$ are two vectors in $\mathbb{R}^3$, the scalar product of $\vec{u}$ and $\vec{v}$ is defined as the real number $\vec{u} \cdot \vec{v} = x \cdot a + y \cdot b + z \cdot c$. The (Euclidean) norm of the vector $\vec{u}$ is then the square root of its dot product with itself, i.e., the positive real number $||\vec{u}|| = \sqrt{x^2 + y^2 + z^2}$. The norm of a vector is an absolute measure of its “magnitude,” which is why, to define the distance between two points, we consider the norm of the vector they define.

Indeed, if we now consider $(x, y, z) = M$ and $(a, b, c) = N$ as two points, the vector representing the “displacement” from $M$ to $N$ is the vector $\overrightarrow{MN} = (a – x, b – y, c – z)$. Thus, the distance between $M$ and $N$ is defined as the norm of $\overrightarrow{MN}$, i.e., the positive real number
$$d(M, N) = ||\overrightarrow{MN}|| = \sqrt{(a – x)^2 + (b – y)^2 + (c – z)^2}.$$
And, consistent with intuition, swapping the roles of $M$ and $N$ gives the same distance, since
$$d(N, M) = ||\overrightarrow{NM}|| = \sqrt{(x – a)^2 + (y – b)^2 + (z – c)^2}.$$

2.2. Recovering the Coordinates of a Vector from the Dot Product

It is worth noting that the coordinates of a vector in space can be recovered from its dot product with each of the basis vectors $\vec{i}$, $\vec{j}$, and $\vec{k}$. Indeed, if $\vec{u} = (x, y, z) \in \mathbb{R}^3$, by definition of the dot product we have $\vec{u} \cdot \vec{i} = x \cdot 1 + y \cdot 0 + z \cdot 0 = x$, and similarly, $\vec{u} \cdot \vec{j} = y$ and $\vec{u} \cdot \vec{k} = z$.

This observation might seem trivial, but this method is interesting when a vector in space is given in a way other than its usual coordinates, for instance, via a spherical representation. In such cases, the dot product is given by a trigonometric formula, which can then be used to determine the coordinates of the vector.