Time remaining:
representations of lines

label Mathematics
account_circle Unassigned
schedule 1 Day
account_balance_wallet $5

Oct 19th, 2017

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

Did you know? You can earn $20 for every friend you invite to Studypool!
Click here to
Refer a Friend
Oct 19th, 2017
Oct 19th, 2017
Oct 20th, 2017
Mark as Final Answer
Unmark as Final Answer
Final Answer

Secure Information

Content will be erased after question is completed.

Final Answer