General Form of a Line

Date: 5/19/96 at 23:7:0
From: Michael Allman
Subject: general form of a line

Dr. Math,

Pertaining to analytic geometry:

Could you please tell me what the equation is for the general form of 
a line in three dimensions? in n dimensions?

And what would be a test for collinearity of points in three 
dimensions? in n dimensions?

I am familiar with matrices and determinants and the like if you want 
to use them.

	Thank you very much.

Michael Allman

Date: 6/14/96 at 9:23:36
From: Doctor Luis
Subject: Re: general form of a line

Well, the general form of a line in n-dimensional space is 
characterized by a parametrization:  that is,  x_i = a_i + u * b_i, 
where u ranges over 
R, and a_i,b_i are constants (i=1,2,3,...,n). This gives n-equations.

If you solve for u in all n-eqs, you obtain the equalities 
(x_1 - a_1)/b_1 = (x_2 - a_2)/b_2 = ... = (x_n - a_n)/b_n  
which determine a line for all ordered n-tuples (x_1,x_2,...,x_n) 
satisfying these equations.    

Maybe looking at the parametric representation sheds more light upon 
the nature of these eqs: that is, for each u in R, u determines an 
n-tuple (x_1,x_2,...,x_n). Now it can be proven that all such n_tuples 
are collinear, but the proof of this I'll leave up to you.  :)  
             
-Doctor Luis,  The Math Forum
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