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/
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