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.

• 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 $y = \rho \ sin (\phi)$.

• 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.

$r$ 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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