Coordinate Axes and Coordinate Planes

To locate a point in space, three mutually perpendicular straight lines are drawn through a fixed point $O$, called the origin. These three lines are the coordinate axes: the $x$-axis (often written $OX$), the $y$-axis ($OY$), and the $z$-axis ($OZ$), each perpendicular to the other two. Positive directions are chosen along each axis, with the opposite ray on each axis taken as negative.

Right-handed system. The three axes are always set up so that they form a right-handed system: if the fingers of the right hand curl from the positive $x$-axis toward the positive $y$-axis, the thumb points along the positive $z$-axis. This convention is fixed throughout coordinate geometry precisely so that formulas such as the cross product (used later for the shortest distance between two lines, Section 7) give consistent, unambiguous signs.

Coordinate planes. Every pair of coordinate axes determines a coordinate plane, containing both axes of the pair:

• The $XY$-plane (or $XOY$-plane) contains the $x$-axis and $y$-axis; every point on it has $z=0$.
• The $YZ$-plane (or $YOZ$-plane) contains the $y$-axis and $z$-axis; every point on it has $x=0$.
• The $ZX$-plane (or $ZOX$-plane) contains the $z$-axis and $x$-axis; every point on it has $y=0$.

These three planes divide the whole of space into eight octants, exactly as the two coordinate axes of a plane divide it into four quadrants. The octant containing all points with $x>0,\ y>0,\ z>0$ is called the first octant, denoted $OXYZ$; the remaining seven octants are obtained by allowing one, two, or all three coordinates to be negative.

Reading the sign of a coordinate. For any point $P(x,y,z)$ in space: $x$ is the (signed) perpendicular distance of $P$ from the $YZ$-plane, $y$ is the perpendicular distance of $P$ from the $ZX$-plane, and $z$ is the perpendicular distance of $P$ from the $XY$-plane -- positive if $P$ lies on the same side as the positive axis, negative otherwise. Equivalently, $x$, $y$, $z$ are the signed lengths of the projections of $OP$ onto the $x$-, $y$- and $z$-axes respectively. This is the natural three-dimensional extension of how the sign of a 2D coordinate is read off from which side of an axis a point lies. …