Equation of a Line in Three or More Dimensions

Date: 05/18/2000 at 22:15:00
From: Paul Jarosch
Subject: 3D+ Line Equation

I know that (in two dimensions) the equation of a line is y = mx + b, 
where m = (rise/run). But what would it be in three dimensions? What 
about four or more dimensions? Can this be solved using vectors or 
trigonometry?

- Paul

Date: 05/19/2000 at 12:45:04
From: Doctor Rob
Subject: Re: 3D+ Line Equation

Thanks for writing to Ask Dr. Math, Paul.

A linear equation (that is, one whose degree in the variables is 1) 
represents a plane in 3D, and a hyperplane in 4D, 5D, etc. A line in n 
dimensions is given as the intersection of n - 1 of these, so doesn't 
have a single equation, but a set of n - 1 simultaneous equations. For 
example, the x-axis has the equations y = 0, z = 0, in the 
three-dimensional Cartesian xyz-coordinate system.

Vectors can be useful, since the equation of a plane perpendicular to 
the vector (a,b,c) in 3-space is (a,b,c).(x,y,z) = d,  where "." means 
dot-product. Then a vector along a line in n-space is one that is 
perpendicular to n - 1 vectors, that is, whose dot product with two 
given constant vectors is zero. Thus the equations of a line take the 
form:

     X.V(1)   = D(1),
     X.V(2)   = D(2),
       :
     X.V(n-1) = D(n-1).

Here X and each V(i) are n-long vectors. The components of X are the n 
variables, and each V(i) is a constant vector. These can be 
consolidated into a matrix form X.V = D, where V is an n-by-(n-1) 
rectangular matrix whose columns are V(1), ..., V(n-1), and D is an 
(n-1)-long vector whose components are D(1), ..., D(n-1).

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/