The Three-Dimensional Coordinate System
Simple Explanation
The familiar 2D (x,y) coordinate plane extends to three dimensions by adding a third axis, z, perpendicular to both x and y. Every point in 3D space is then located by an ordered triple (x, y, z), giving its position along each of the three axes.
Why Do We Need It?
This is the essential foundation for describing any real three-dimensional object or location mathematically — the line, plane, and sphere later in this chapter are all built directly on this coordinate system.
See It
A sketch of three perpendicular axes labelled x, y, and z meeting at the origin, with a point P marked off to the side representing the location (3,4,2)
Worked Example
Locate a point in 3D space
Describe how to reach the point P(3,4,2) starting from the origin.
Why Does This Work?
Just as two perpendicular axes (x and y) are enough to uniquely locate any point in a flat plane, a third axis (z), perpendicular to both of the first two, adds exactly the one additional degree of freedom needed to locate any point in the space around us — no more, no fewer.
Real-Life Example
GPS coordinates with altitude
A GPS system needs to describe not just a location on the ground, but also its height above sea level.
Latitude and longitude act like x and y, and altitude acts like the third (z) coordinate — together giving a full three-dimensional location.
Practice
Which ordered triple describes a point 5 units along the x-axis, 0 along y, and 2 along z?
EasyCommon mistake
Forgetting that the ORDER of the coordinates matters — (3,4,2) and (4,3,2) describe different points, exactly like in two dimensions.
Quick Review
- Every point in 3D space is described by an ordered triple (x, y, z).
- The z-axis is perpendicular to both the x-axis and the y-axis.
- Order matters: (x,y,z) is not the same as any rearrangement of its values.