### Theory:

Definition:
If $$A$$, $$B$$ and $$C$$ are three non-empty sets then the cartesian product of three sets is the set of all possible ordered triples given by $$A \times B \times C$$ $$=$$ $$\{(a, b, c) \text{ for all } a \in A, b \in B, c \in C\}$$.
Illustration:
Let us consider the following example for the geometrical understanding of cartesian product of two and three sets.

Consider three sets $$A$$, $$B$$ and $$C$$.

Where $$A$$ $$=$$ $$\{2, 3\}$$, $$B$$ $$=$$ $$\{2, 3\}$$ and $$C$$ $$=$$ $$\{2, 3\}$$.

First, let us find the product $$A \times B$$.

$$A \times B$$ $$=$$ $$\{2, 3\} \times \{2, 3\}$$

$$A \times B$$ $$=$$ $$\{(2 ,2), (2, 3), (3, 2), (3, 3)\}$$

The product $$A \times B$$ is geometrically represented as follows:

Here, the product of two sets is represented in the $$xy$$ $$-$$ plane.

The product $$A \times B$$ represents the vertices of a square in two dimensions.

Now, let us find the product $$A \times B \times C$$.

$$A \times B \times C$$ $$=$$ $$(A \times B) \times C$$

$$A \times B \times C$$ $$=$$ $$\{(2 ,2), (2, 3), (3, 2), (3, 3)\}$$ $$\times$$ $$\{2, 3\}$$

$$A \times B \times C$$ $$=$$ $$\{(2 ,2, 2), (2 ,2, 3), (2, 3, 2), (2, 3, 3), (3, 2, 2), (3, 2, 3), (3, 3, 2), (3, 3, 3)\}$$.

The product $$A \times B \times C$$ is geometrically represented as follows:

Here, the product of three sets is represented in the $$xyz$$ $$-$$ plane.

The product $$A \times B \times C$$ represents the vertices of a cube in three dimensions.

Important!
The cartesian product of two non-empty sets provides a shape in $$2$$ $$-$$ dimension, whereas the cartesian product of three non-empty sets provides a shape in $$3$$ $$-$$ dimension.