**Introduction**

** ** ** ** Coordinates which are applied in three
dimensions is called three dimensional coordinates. It requires three different numbers to locate the position of a point in the space.

The basic formula for the distance between any two Points A (x_{1,} y_{1}, z_{1}) and B (x_{2,} y_{2}, z_{2}) in
three dimensional coordinates is given by

** √ ( (x _{1} - x_{2})^{2} + (y_{1} -
y_{2)2 +} (z_{1} - z_{2})^{2} )**

**Parametric Form of a Straight Line:**

The straight line equations which are passing through the point (x1, y1, z1) can be expressed in the form of

x = at +
x_{1 }

_{ } y = bt + y_{1}

_{ } z = ct
+ z_{1}

where ‘t’ is a parameter and a, b, c are directional vectors.

**Symmetric Form of a Straight Line:**

The straight line equation which are passing through the point (x_{1,} y_{1}, z_{1}) can be expressed in the form of

` ( x - x1 ) / a` = `(y - y1)/b` = `( z - z1 ) / c`

where a, b, c are directional vectors.

The equation of the line joining the points A (x_{1,} y_{1}, z_{1}) and B (x_{2,} y_{2}, z_{2}) is given by

** **` ( x - x1) / ( x2 - x1)` **= `( y - y1 ) / ( y2 - y1 )` = `( z - z1 ) /( z2 - z1
)`**

The example problems are solved below for three dimensional coordinates.

**1) Find the distance between the points (2, 3, 4) and (4, 6, 8).**

**Sol**:

The basic formula to solve the distance between any two points is given as

d = √ ( (x_{1} - x_{2})^{2} + (y_{1} - y_{2)2 +} (z_{1} - z_{2})^{2} )

**
=** **√** ( (2 - 4)^{2} + (3 - 6_{)2 +} (4 - 8)^{2} ) ** **

**
= ** **√** ( (-2)^{2} + (-3)_{2 +} (-4^{2}) ** **

**
=** **√** ( 4 + 9 + 16)

**
=** √ 29

**
=** 5.38

**2) Find the equation of the straight line joining the points (2, 0, 3) and (4, -1, 2).**

**Sol:**

** ** The equation of the line is given by

** ** `( x - x1) / ( x2 - x1)` = `( y - y1 ) / ( y2 - y1 )
` = `( z - z1 ) /( z2 - z1 )`

`( x - 2) / ( 4 - 2)` = `(y - 0 ) / ( -1 - 0 )` =` ( z - 3 ) /( 2 - 3 )`

` ( x - 2) /2` = `y / (-1)` = `( z - 3 ) / (-1)`