Mathematics · Ch 5 — Introduction to Three-Dimensional Geometry

Coordinate Axes and Coordinate Planes in Three Dimensional Space

5.2

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:

Along the $z$ -axis: Distances measured upwards from the $XY$ -plane, in the direction of $OZ$ , are taken as positive. Distances measured downwards from the $XY$ -plane, in the direction of $OZ'$ , are taken as negative.
Along the $y$ -axis: Distances measured to the right of the $ZX$ -plane, along $OY$ , are positive. Distances to the left of the $ZX$ -plane, along $OY'$ , are negative.
Along the $x$ -axis: Distances measured in front of the $YZ$ -plane, along $OX$ , are positive. Distances to the back of the $YZ$ -plane, along $OX'$ , are negative.

Watch out

A common confusion is mixing up which axis is "vertical." In the standard setup, the $z$ -axis is vertical, not the $y$ -axis. The $XY$ -plane is the horizontal "floor" plane.

The Eight Octants

The three coordinate planes — $XY$ , $YZ$ , and $ZX$ — divide all of space into eight regions. Each region is called an octant. These octants are analogous to the four quadrants in two-dimensional geometry.

The eight octants are named by listing which side of each coordinate plane the region lies on. The notation uses the axes directions: $X$ (positive $x$ ), $X'$ (negative $x$ ), $Y$ (positive $y$ ), $Y'$ (negative $y$ ), $Z$ (positive $z$ ), $Z'$ (negative $z$ ). The eight octants are:

Octant	Name	Sign of $x$	Sign of $y$	Sign of $z$
I	$XOYZ$	$+$	$+$	$+$

Figure 11.1Three mutually perpendicular coordinate planes meeting at the origin O

Fig. 11.1 — Three mutually perpendicular coordinate planes meeting at the origin O

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

The figure shows the three-dimensional coordinate system built from three mutually perpendicular planes. At the centre is the origin O, where all three planes meet. Three double-headed axes radiate from O: the vertical axis is labelled Z′OZ (Z up, Z′ down), the horizontal axis is Y′OY (Y to the right, Y′ to the left), and the diagonal axis is X′OX (X′ in the upper-right direction, X in the lower-left). The three light-blue parallelograms represent the coordinate planes: the horizontal XOY plane (the plane of the paper), the upright YOZ plane, and the slanted ZOX plane. Together, these form the rectangular coordinate system.

The physical idea is that any point in space can be located by its perpendicular distances from these three planes. The XOY plane is taken as horizontal, with the Z-axis vertical. Distances measured upward along OZ are positive, downward along OZ′ are negative. Distances to the right of the ZX-plane along OY are positive, to the left along OY′ are negative. Distances in front of the YZ-plane along OX are positive, to the back along OX′ are negative. The three planes divide all of space into eight octants, each named by a combination of the positive or negative directions of the three axes.

Important

The key formula the textbook develops from this figure is the distance between two points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$ in three-dimensional space:

$PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$

This is a direct extension of the Pythagorean theorem to three dimensions. Each symbol represents a coordinate along the corresponding axis: $x$ along X′OX, $y$ along Y′OY, and $z$ along Z′OZ.

The figure also leads to the section formula for internal and external division in 3D. If a point $R$ divides the line segment joining $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$ in the ratio $m:n$ , then the coordinates of $R$ are: …

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