INTRODUCTION : BASIC FORMULA
we need three nonco planar directions as our reference axis to specify the position of any point. It is most convenient to take the three noncoplanar directions (denoted conventionally by , and directions) as being mutually perpendicular to each other.
Such a reference axes is termed a rectangular coordinate axis. Any point in space can now be specified with respect to this frame by specifying the components (coordinates) of along each axis.
For example, in the figure below,
the coordinates of along the , and axis are given to be , and . Thus, can be specified as
In two dimensions, two coordinates were sufficient to uniquely determine any point; in three dimensions, we need three coordinates.
Note that a two dimensional coordinate axes divides the plane into four quadrants; a three dimensional coordinate axes will divide the space into eight “compartments” known as octants. As an elementary exercise, write down the coordinates of some arbitrary points in each of the eight octants.
DISTANCE FORMULA
Let and be two arbitrary points. We need to find the distance between and in terms of the coordinates of and . Observe the following diagram carefully:
is simply the length of the diagonal of the cuboid drawn (as depicted above):
This is the distance formula for three dimensions. It has a form exactly similar to the distance formula for two dimensions.
As an example, let and We have,
Incidentally, is the mid point of since and .
Example: 1  
Find the locus of the point equidistant from the four points and

Solution: 1  
Let We have,
Similarly, and Thus, the point P is

Example: 2  
Find the locus of a point which moves so that its distances from the points and are always equal.

Solution: 2  
will obviously lie on the perpendicular bisector of . Let the coordinates of be .
Therefore,
This is the required locus of (if you think carefully, it is a plane)
