The space in which we live in, is three-dimensional. In general terms, a dimension means a direction of movement of an object or a person. The 3 dimensions are :
I) moving or ,
II) moving or and
III) moving or
In the terminology of mathematics, the dimensions become a necessary (and sufficient) set of coordinates, required to uniquely specify the location of an object/ a point w.r.t a reference. The reference is known as the origin or pole.
For a 3D space, total number of coordinates required to specify a location is .
We have different coordinate systems, which specify location of a point w.r.t. origin.
Rectangular or Cartesian Coordinate System
The most intuitive coordinate system is the Cartesian Coordinate system. There are 3 mutually perpendicular axes, viz. , and . Their point of intersection is the origin. Each axis is a real number line, with origin indicating the number . Each coordinate of a point indicates the directed distance of the point from the origin, measured along respective axis. The directed distance can be negative or positive or 0.
The figure below is self-explanatory.
Cylindrical Polar Coordinate System
A slightly advanced version is the cylindrical polar coordinate system. The coordinate is as it is in the Cartesian coordinate system. Remaining 2 coordinates are given by :
Alternatively, and .
Spherical Polar Coordinate System
A common observation is, almost all astronomical bodies are spherical (including earth). It is inconvenient to use the rectangular / cylindrical systems to specify the trajectory of objects moving along surface of such bodies. Hence, a new system has been introduced. It uses 2 angles and a distance as the coordinates.
is the distance of the point from the origin. The angle made by this line with positive axis is the angle . The other angle is , which is the angle made by the projection of point on plane with positive axis.
From the figure,
Note that, although can take values from to , to specify a point in 3D space, it is enough to vary it from to .
Distance and Section Formulas
Let and be any 2 points. The distance between them is given by
If divides in the ratio , then
sign, when divides internally, sign, when divides externally.
Note : Vectors are such an important tool in 3D geometry, that the analysis becomes way too simple!