Coordinate Axes and Coordinate Planes in Three Dimensional Space

The Three Mutually Perpendicular Planes

We begin with a point $O$ in space. Through $O$, imagine three planes that are all mutually perpendicular — each plane meets the other two at right angles. These three planes intersect each other along three lines that also pass through $O$. Those lines are called the coordinate axes.

The line formed by the intersection of the first two planes is called the $x$-axis, labelled $X'OX$. The line from the intersection of the second and third planes is the $y$-axis, labelled $Y'OY$. The third line, from the intersection of the first and third planes, is the $z$-axis, labelled $Z'OZ$. Because the planes are mutually perpendicular, these three axes are also mutually perpendicular to each other. Together, they form the rectangular coordinate system in three-dimensional space.

The point $O$ where all three axes meet is called the origin.

The Three Coordinate Planes

Each pair of axes determines a plane. These are the three coordinate planes:

• The plane containing the $x$-axis and the $y$-axis is the $XY$-plane (also written as the $XOY$ plane).
• The plane containing the $y$-axis and the $z$-axis is the $YZ$-plane (the $YOZ$ plane).
• The plane containing the $z$-axis and the $x$-axis is the $ZX$-plane (the $ZOX$ plane).

Each coordinate plane is perpendicular to the remaining axis. For instance, the $XY$-plane is perpendicular to the $z$-axis.

Orientation and Sign Conventions

We take the $XOY$ plane (the $XY$-plane) as the plane of the paper. The line $Z'OZ$ (the $z$-axis) is then perpendicular to this plane. If we think of the plane of the paper as horizontal, the $z$-axis is vertical.

Distances are measured with a consistent sign convention: …