Rectangular coordinates in $R^3$

Calculus for functions $z = f(x,\,y)$ of two (or more) variables relies heavily on what you already know about the calculus of functions $y = f(y)$ of one variable. Since the graph of a function $z = f(x,\, y)$ of a function of two variables is a surface in $3$-space we'll have to introduce suitable coordinate systems in $3$-space. The first one is the Cartesian $xyz$-coordinate system obtained by adding the $z$-axis perpendicular to the usual $xy$-coordinate system in the $xy$-plane.

Example 1: find the distance of the point $P(5,\,4,\, 7)$ from the plane $z = 3$. Solution: the plane $z = 3$ shown in green is parallel to the plane $z = 0$, in other words to the $xy$-plane. It consists of all points with coordinates $(x,\,y,\, 3)$. So its graph is the one shown to the right, and $P$ lies above the plane $z = 3$. The distance of $P(5,\,4,\,7)$ from the plane $z = 3$ is the length of the dark blue line. Thus $$\hbox{distance}\{(5,\, 4,\, 7), \, z=3\} \ = \ 7 - 3 \ = \ 4 \,.$$