The three-dimensional coordinate system: the right-hand rule, the coordinate planes and plotting points
A point on a line needs one number, a point in a plane needs two, and a point in space needs three. Space coordinates use three axes x, y and z that are perpendicular to each other and meet at the origin O (in Arabic النظام الإحداثي المتعامد). Their directions follow the right-hand rule: curl the fingers of your right hand from the positive x-axis towards the positive y-axis and your thumb points along the positive z-axis. In our pictures x runs to the right, y goes away from us and z points up. The three planes through pairs of axes are the coordinate planes: the xy-plane has equation z = 0, the xz-plane has y = 0 and the yz-plane has x = 0. To plot P(x, y, z) go to the point (x, y) of the xy-plane first, then climb z units up (or |z| down if z is negative). Equivalently, P is the far corner of a box with one corner at O and edges of length |x|, |y| and |z| along the axes. This gives quick distances. P is |z| from the xy-plane, |y| from the xz-plane and |x| from the yz-plane, and it is √(y² + z²) from the x-axis (the diagonal of one face of the box).
| point | where it lies |
|---|---|
| A(5, 2, 0) | in the xy-plane (z = 0) |
| B(0, −3, 4) | in the yz-plane (x = 0) |
| C(0, 0, −6) | on the z-axis (x = 0 and y = 0) |
| D(−2, 4, 1) | in no coordinate plane |
Use a corner. Look at the corner where two walls and the floor of your room meet. Say which wall is the xz-plane, which is the yz-plane and which axis is the z-axis. Check with the right-hand rule that x, y, z are in the right order.