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representations of lines

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Apr 9th, 2015

Representations of a Line in Two and Three Dimensions

Two points P1 and Pon a line, L, determine L.

L can be described parametrically as the set of points with coordinates those of P+ s * (P2 - P1) for some number s.

(P2 - P1) is a vector which points in the direction of L.

In two dimensions so that the vectors here are 2-vectors, there is only one direction perpendicular to L, and that direction can be obtained by switching the coordinates of (P2- P1 and changing one sign, (thus (7, -4) is perpendicular to (4, 7)).
With N the perpendicular vector, the equation of the line becomes N[img src="http://ocw.mit.edu/ans7870/18/18.013a/textbook/HTML/symbols/dot.gif" width="10" height="13" align="absmiddle">r = NP1.

We do this out explicitly  L consists of the points obeying

x  = P1x  + s * (P2x - P1x )

y  = P1y  + s * (P2y - P1y )

and the equation for L is

(P2y - P1y ) x - (P2x - P1x )y = (P2y - P1y )P1x - (P2x - P1x )P1y

which when solved for y is

[img src="http://ocw.mit.edu/ans7870/18/18.013a/textbook/HTML/chapter05/equations/sections_eqn01.gif" width="145" height="50" align="absmiddle"> for some constant C.

The ratio [img src="http://ocw.mit.edu/ans7870/18/18.013a/textbook/HTML/chapter05/equations/sections_eqn02.gif" width="85" height="50" align="absmiddle">, the coefficient of x in the equation for the line, is the difference of y coordinates of the two points divided by the difference in x coordinates. It is called the slope of the line L.

Apr 9th, 2015

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