Coordinates in space

On a flat page a point needs two numbers (x, y): its distances from two perpendicular axes. A point floating in a room needs three, because space has three directions: across, along and up.

So we use three axes at right angles, the x-, y- and z-axes, meeting at the origin O. Each pair of axes makes a flat coordinate plane:

• The XY-plane (the floor), the YZ-plane and the ZX-plane (two walls).
• Picture the corner of a room: two walls and the floor meet at one corner, and their edges are the three axes.

A point P is written (x, y, z):

• x = its distance from the YZ-plane, y = its distance from the ZX-plane, z = its distance from the XY-plane, each with a sign showing which side it is on.
• P(3, –2, 6) is 3 units from the YZ-plane, 2 from the ZX-plane (on the negative side) and 6 above the XY-plane. The foot of its perpendicular on the floor is (3, –2, 0).

Points on the planes and axes. If a point is in a plane, its distance from that plane is zero:

• In the XY-plane z = 0, so the point is (x, y, 0). In the YZ-plane x = 0: (0, y, z). In the ZX-plane y = 0: (x, 0, z).
• On the x-axis the point is in two planes at once, so two coordinates are 0: (x, 0, 0). Likewise (0, y, 0) on the y-axis and (0, 0, z) on the z-axis.
• So (0, 4, 0) is on the y-axis, (3, 0, –2) is in the ZX-plane and (0, 0, 7) is on the z-axis.

Octants. The three planes cut space into eight parts called octants, just as two axes cut a page into four quadrants. Think of a two-storey house with four rooms on each floor: the upper floor has z positive, the lower floor z negative.

A quick way to find the octant: use x and y to get the quadrant (1 to 4) as on a flat page. If z is positive, that is the octant; if z is negative, add 4.

• (2, –3, 4): x +, y – is quadrant 4, z positive: octant IV.
• (–1, –2, –3): quadrant 3, z negative: 3 + 4 = octant VII.
• (5, 1, –2): quadrant 1, z negative: octant V.