Three Dimensional Coordinate Geometry I

  • Coordinate Systems

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 right or left,

II) moving forward or backward and

III) moving up or down

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 3.

We have different coordinate systems, which specify location of a point w.r.t. origin.

  • Rectangular or Cartesian Coordinate System (x,y,z)

The most intuitive coordinate system is the Cartesian Coordinate system. There are 3 mutually perpendicular axes, viz. X, Y and Z. Their point of intersection is the origin. Each axis is a real number line, with origin indicating the number 0. 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.

cartesian.jpg

 

  • Cylindrical Polar Coordinate System (\rho, \phi, z)

A slightly advanced version is the cylindrical polar coordinate system. The z coordinate is as it is in the Cartesian coordinate system. Remaining 2 coordinates are given by :

{\rho = \sqrt {x^2 + y^2}, \ \phi = \tan^{-1} \Big(\frac y x \Big)}

Alternatively, x = \rho \ cos (\phi) and y = \rho \ sin (\phi).

cyl_polar.jpg

 

 

  • Spherical Polar Coordinate System (r, \theta, \phi)

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.

sph_polar.jpg

 

r is the distance of the point from the origin. The angle made by this line with positive Z axis is the angle \theta. The other angle is \phi, which is the angle made by the projection of point on XY plane with positive X axis.

From the figure,

{z = r \ cos (\theta), \ x = r \ sin (\theta) cos (\phi), \ y = r \ sin (\theta) sin (\phi)}

Note that, although \theta can take values from 0 to 2 \pi, to specify a point in 3D space, it is enough to vary it from 0 to \pi.

  • Distance and Section Formulas

Let P (x_1,y_1,z_1) and Q(x_2,y_2,z_2) be any 2 points. The distance between them is given by

{\sqrt {(x_1-x_2)^2 + (y_1 - y_2)^2+ (z_1 - z_2)^2}}

If R(x_3,y_3,z_3) divides PQ in the ratio m:n, then

{x_3 = \frac {mx_2 \pm nx_1}{m \pm n}, y_3 = \frac {my_2 \pm ny_1}{m\pm n}, z_3 = \frac {mz_2 \pm nz_1}{m \pm n}}

+ sign, when R divides PQ internally, - sign, when R divides PQ externally.

 

Note : Vectors are such an important tool in 3D geometry, that the analysis becomes way too simple!

 

 

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