Representations of a Line in Two and Three Dimensions
Two points P1 and P2 on a line, L, determine L.
L can be described parametrically as the set of points with coordinates those of P1 + 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.
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