Locating Points in Three-Dimensional Space

Why Two Numbers Are Sometimes Not Enough

Two coordinates locate a point on a flat surface, but they cannot distinguish points at different heights above that surface. Imagine two lamps hanging directly above the same spot on the floor at different heights. Their floor positions agree, yet their positions in the room are different. A third coordinate records the missing information.

A room provides a useful model. Choose a corner where the floor and two adjacent walls meet. The floor and the two walls are three mutually perpendicular planes: each meets the others at a right angle. To locate the tip of a hanging bulb or the centre of a ceiling fan, we need its position relative to both walls and its height above the floor. A flying aeroplane or a ball moving through the air likewise has a location that cannot always be described on one flat plane.

The word dimension describes an independent direction needed to specify location. A point on a line needs one coordinate; a point in a plane needs two; a point in ordinary space needs three. Our axes provide these directions, while the chosen origin gives the common starting point. Once the system is fixed, the same point always has the same three coordinates in that system.

Axes, Origin and Coordinate Planes

Extend the two-dimensional coordinate system by adding a third axis perpendicular to both of the original axes. The three axes meet at one point, the origin. Each pair of axes determines a coordinate plane, giving us three reference surfaces for locating points.

The x- and y-axes determine the XY-plane. The y- and z-axes determine the YZ-plane. The z- and x-axes determine the ZX-plane, also called the XZ-plane. A plane is a flat surface that extends without bound; the shaded rectangles in a diagram represent only small portions of these planes. All three planes meet at O, and each pair intersects along an axis.

A drawing on paper shows these axes in perspective, so they may not appear to meet at right angles on the page. Their actual spatial directions are perpendicular. In the diagrams here, positive x points diagonally down-left, positive y points right, and positive z points upward. These are drawing choices: turning the picture does not change the coordinate relationships.

Each axis has a positive and a negative half. Along z, positive points may be described as above the XY-plane and negative points as below it. Along y, the sign tells which side of the ZX-plane the point occupies. Along x, it tells which side of the YZ-plane the point occupies. Use the labelled positive directions rather than assuming that right or upward always means the same coordinate in every picture.

An Ordered Triple Is a Location

The notation P(x,y,z) names a point with three coordinates in a fixed order. The first refers to the x-direction, the second to y and the third to z. The coordinates are signed numbers, so they describe both how far and in which direction to move from the reference planes.

To construct P=(2,4,5), first move two units along the positive x-axis to L=(2,0,0). From L move four units parallel to the positive y-axis to M=(2,4,0), which lies in the XY-plane. Then move five units parallel to positive z to reach P. These are three perpendicular coordinate movements; their total length is not the direct distance OP.

Conversely, drop a perpendicular from P to the XY-plane. Its foot is M=(x,y,0), because the perpendicular movement changes only z. From M drop a perpendicular to the x-axis, reaching L=(x,0,0). The signed coordinate changes OL, LM and MP correspond to x,y,z. When a coordinate is negative, the movement is in the opposite direction, while the length of the movement is the coordinate’s absolute value.

Every ordered triple gives exactly one point by this construction, and every point gives exactly one triple. This is a one-to-one correspondence: no point has two different triples in the same fixed system, and no triple identifies two different points. The coordinates do depend on the chosen origin, directions and scales, so changing the coordinate system can change a point’s description.

The Same Point from Three Parallel Planes

There is another way to picture a point: use three planes parallel to the coordinate planes. Each fixes one coordinate. Their common intersection fixes all three at once, so it is the same point found by the successive-movement construction.

For P=(2,4,5), imagine the plane through (2,0,0) parallel to the YZ-plane, the plane through (0,4,0) parallel to the ZX-plane, and the plane through (0,0,5) parallel to the XY-plane. They contain points with x=2, y=4 and z=5 respectively. Their single common point is P. This is a way to understand coordinates; it does not require a general theory of plane equations.

Together with the coordinate planes, these parallel planes outline a rectangular box for positive coordinates. The corners share different choices of zero or the corresponding coordinate. For instance, the corner reached by keeping x=2 and z=5 but setting y to zero is F=(2,0,5). Its position is on the ZX-plane.

Octants and Boundary Points

In a plane, two axes divide the plane into four quadrants. In space, three coordinate planes divide the space into eight open regions called octants. Each octant has a distinct combination of signs, because each of the three coordinates can be positive or negative.

Use the chapter’s numbering convention. The first four octants have positive z and follow the usual order of x,y signs in quadrants I–IV. The next four repeat those x,y signs with negative z. The table is a sign map, so it works even when a perspective drawing is hard to interpret.

A zero coordinate means that the point lies on a coordinate plane. Two zero coordinates put the point on an axis. Three zeros give the origin. Such points are not inside any open octant, so assigning them a positive or negative sign for the zero coordinate would invent information.