In a plane the position of a point is determined by an ordered pair (x, y) of real numbers obtained with reference to two straight lines in the plane generally at right angles. The position of a point in space is, however, determined by an ordered triad (x, y, z) of real numbers. We now proceed to explain as to how this is done.

Co-ordinates of a Point in Space


Let X¢OX, Z¢OZ be two perpendicular straight lines determining the XOZ plane. Through O, their point of intersection, called the origin, draw the line Y¢OY perpendicular to the XOZplane so that we have threemutually perpendicular straight lines X¢OX, Y¢OY, Z¢OZ known as rectangular co-ordinate axes. These three axes, taken in pairs, determine the three planes, XOY, YOZ and ZOX or briefly the XY, YZ, ZX planes mutually at right angles, known as rectangular co-ordinate planes.

Through any point P, in space, draw three planes parallel to the three co-ordinate planes (being also perpendicular to the corresponding axes) to main axes in A, B, C.

Let, OA = x, OB = y and OC = z. These three numbers x, y, z taken in this order determined by the point P, are called the co-ordinates of the point P. We refer to the ordered triad (x, y, z) formed of the co-ordinates of the point P as the point P itself.

Any one of these x, y, z will be positive or negative according as it is measured from O, along the corresponding axis, in the positive or the negative direction.

Conversely, given an ordered triad (x, y, z) of numbers, we can find the point whose co-ordinates are x, y, z. To do this, we proceed as follows:

(i)        Measure OA, OB, OC along OX, OY, OZ equals to x, y, z respectively.

(ii)    Through the points A, B, C draw planes parallel to the co-ordinate planes YZ, ZX, XY respectively. The point of intersection of these planes is the required point P.