General Form of a LineDate: 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 Check out our web site! |
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