Lines in Space      (Click here to skip to Planes in Space)    

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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In a Nut Shell:   There are three types of lines in space.  Those that are parallel, those

that intersect, and those that are skew (neither parallel nor intersect).   The equation of

a line in space can be found by vector addition as described below.

 

Let    r  =  < x, y, z >  be a vector from the origin, O,  to an arbitrary point   P(x, y, z)

on a line, L,  in space.   Let  ro  =  < xo, yo, zo >   be a vector from the origin to the

point  Po (xo, yo, zo)  on the same line, L.    Let  V  be a vector  < a, b, c >   parallel

to the line  L.  Let  t  be a constant parameter.

 

Now      let  tV  be a vector along (or parallel to line L) such that

 

             tV   =       ta i    +  tb j  +  tc k is the vector from    ro    to   r .  

             

Then by vector addition:  (Key step in determining the equation for the line)

 

               r   =  ro    +   tV    (this is the equation of the line in vector form)

 

Equation of the line, L,  in scalar form:  x  =  xo   +   at,   y  =  yo   +   bt,   z  =  zo   +   ct

 

 

The equation of the line, L,  in “symmetric” form is obtained from the scalar form by

solving for t.  The result is:

 

     [x   - xo] /  a      =     [y   - yo] /  b      =         [z   - zo] /  c      =     t   

 

Click here for an example.

 

 

2.

 

Click here for discussion and an example of planes in space.

 


Copyright © 2011 Richard C. Coddington

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