representations of lines
Mathematics

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Representations of a Line in Two and Three Dimensions
Two points P_{1} and P_{2 }on a line, L, determine L.
L can be described parametrically as the set of points with coordinates those of P_{1 }+ s * (P_{2}  P_{1}) for some number s.
(P_{2}  P_{1}) is a vector which points in the direction of L.
In two dimensions so that the vectors here are 2vectors, there is only one direction perpendicular to L, and that direction can be obtained by switching the coordinates of (P_{2} P_{1}) 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 = NP_{1}.
We do this out explicitly L consists of the points obeying
x = P_{1x} + s * (P_{2x}  P_{1x} )
y = P_{1y} + s * (P_{2y}  P_{1y} )
and the equation for L is
(P_{2y}  P_{1y} ) x  (P_{2x}  P_{1x} )y = (P_{2y}  P_{1y} )P_{1x}  (P_{2x}  P_{1x} )P_{1y}
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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