3D Coordinate Systems and Distance Formula

Definition A three-dimensional rectangular coordinate system consists of a point $O$, the origin, and three mutually perpendicular number lines through $O$, the $x$-, $y$- and $z$-axes, each with its zero at $O$ and all with the same unit of length. The axes follow the right-hand rule. When the fingers of your right hand curl from the positive $x$-axis toward the positive $y$-axis, your thumb points along the positive $z$-axis. Each point $P$ in space then corresponds to exactly one ordered triple $(a, b, c)$ of real numbers, its coordinates. Here $a$, $b$ and $c$ are the positions on the $x$-, $y$- and $z$-axes of the feet of the perpendiculars dropped from $P$ to those axes. Conversely, each ordered triple names exactly one point. The set of all ordered triples of real numbers is written $\mathbb{R}^3$.

Definition The coordinate planes are the $xy$-plane, where $z = 0$, the $yz$-plane, where $x = 0$, and the $xz$-plane, where $y = 0$. In space a single equation such as $z = 0$ or $y = 4$ describes a whole plane, not a line. The projection of $P = (a, b, c)$ onto the $xy$-plane is the point $(a, b, 0)$ directly below or above $P$, or $P$ itself when $c = 0$. Likewise its projections onto the $yz$-plane and the $xz$-plane are $(0, b, c)$ and $(a, 0, c)$. The distance from $P$ to the $xy$-plane is $\lvert c \rvert$, to the $yz$-plane it is $\lvert a \rvert$, and to the $xz$-plane it is $\lvert b \rvert$.

Definition The three coordinate planes divide the rest of space into eight regions called octants, one for each choice of signs of the three coordinates. The first octant is the region where $x > 0$, $y > 0$ and $z > 0$. There is no standard numbering of the other seven, so they are described by their signs. A point with a zero coordinate lies on a coordinate plane and belongs to no octant.

Definition The distance between the points $P_1 = (x_1, y_1, z_1)$ and $P_2 = (x_2, y_2, z_2)$ is

Here is why. Let $Q = (x_2, y_2, z_1)$, the point at the height of $P_1$ directly below or above $P_2$. The segment $P_1Q$ lies in the plane $z = z_1$, so the distance formula of the plane gives

The segment $QP_2$ is vertical with length $d(Q, P_2) = \lvert z_2 - z_1 \rvert$, so it is perpendicular to $P_1Q$. The Pythagorean theorem in the right triangle $P_1QP_2$ gives

which is the formula. When $Q$ coincides with $P_1$ or with $P_2$ the formula still holds.

Definition The midpoint of the segment from $P_1 = (x_1, y_1, z_1)$ to $P_2 = (x_2, y_2, z_2)$ is

It lies on the segment and is at the same distance from $P_1$ and $P_2$, namely half of $d(P_1, P_2)$.

Definition The sphere with center $C = (h, k, l)$ and radius $r > 0$ is the set of all points in space whose distance from $C$ is $r$. By the distance formula, a point $(x, y, z)$ lies on it exactly when

This is the standard equation of the sphere. The sphere is only the surface. The points at distance less than $r$ from $C$ lie inside it, and the sphere together with its inside is the solid ball, the set where the left side of this equation is at most $r^2$.

Definition An equation of the form

can be rewritten by completing the square in each variable as

If the right side is positive, the equation describes the sphere with center $(-\frac{D}{2}, -\frac{E}{2}, -\frac{F}{2})$ whose radius is the square root of the right side. If the right side is zero, it describes that single point, and if it is negative, no point at all.