Solve Three Dimensional Coordinates

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 (x1, y1, z1) and B (x2, y2, z2) in three dimensional coordinates is given by

                                                         √ ( (x1 - x2)2 + (y1 - y2)2 + (z1 - z2)2 )

 

Equations of straight lines to solve three dimensional coordinates:

 

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

                            y =  bt + y1

                             z =  ct + z1

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 (x1, y1, z1) 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 (x1, y1, z1) and B (x2, y2, z2) 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.

 

Example problems to solve 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 = √ ( (x1 - x2)2 + (y1 - y2)2 + (z1 - z2)2 )

                                        = ( (2 - 4)2 + (3 - 6)2 + (4 - 8)2    

                                        =  ( (-2)2 + (-3)2 + (-42 

                                        = ( 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)`