#[Coordinates of points in three dimensions][Coordenadas de puntos en tres dimensiones]#

We live in what we perceive to be three dimensional space, meaning that we perceive three independent directions in which we might imagine moving; for instance:

• Forward/backwards, or the $x$-direction, where forward is counted as positive,
• Left/right, or the $y$-direction, where left is counted as positive,
• Up/down, or the $z$-direction, where up is counted as positive.

This way, we can assign three coordinates $(x, y, z)$ to each point $P$ in space by noting how much we need to move in these three directions to reach it. For instance, if can reach it by moving 2 units forward, 3 units right, and one unit down, then its ooordinates would be $(2,-3,-1).$ Similarly, we can reach the point $(-4,5,6)$ by moving 4 units backwards, 5 units left, then 6 units up. This idea leads to the formal definition of three-dimensional space:

#[Three dimensional space is the set of all triples of real numbers $(x,y,z),$ called points.][El espacio tridimensional es el conjunto de todos los triples de números reales $(x,y,z),$ llamados puntos.]#