Coordinates of a point in space

Given a point P, drop the perpendicular PM to the XY-plane, then from its foot M drop the perpendicular ML to the x-axis. With OL = x, LM = y and MP = z, the ordered triplet (x, y, z) gives the coordinates of P.

Run it backwards to plot a given triplet: mark L at x on the x-axis, move parallel to the y-axis to M, whose plane coordinates are (x, y), then rise (or sink) z perpendicular to the XY-plane to reach P.

Each point of space gives exactly one ordered triplet and each triplet gives exactly one point: the correspondence is one to one.

A second view: through P draw three planes parallel to the coordinate planes. They cut the axes at A, B and C, and OA = x, OB = y, OC = z. Together with the coordinate planes they box in a cuboid with O and P at opposite corners.

Hence x, y and z are the perpendicular distances of P from the YZ, ZX and XY-planes respectively, with signs. x is measured from the YZ-plane, not from the x-axis.

If P lies in octant I all three coordinates are positive; in any other octant the signs change as the octant table shows.