Analytical GeometryDate: 04/11/2002 at 09:49:01 From: R. Kesavan Subject: Analytical Geometry Dear Sir, When two straight lines meet each other perpendicularly, the product of their slopes is -1. However, x and y axes meet at 90 degrees, but the product of their slopes is not -1 because slope of x axis = 0 slope of y axis = infinity. Why? Thanks. R. Kesavan Date: 04/11/2002 at 12:34:16 From: Doctor Peterson Subject: Re: Analytical Geometry Hi, R. This is true because the y axis has an undefined slope, so you can't multiply it by anything. You can only talk about the product of the slopes when both slopes exist, so your statement really needs that qualification. Now, if we pretend infinity is a number, we find that 0*infinity is indeterminate, and -1 is one of the possible values it can be considered to have, so that it does not contradict the theorem; but it has no defined value. You can read about that concept in the Dr. Math FAQ: Dividing by Zero http://mathforum.org/dr.math/faq/faq.divideby0.html But you raise an interesting point: is there a form of this fact that applies to all pairs of lines, including those with infinite slope? The best equation to use for a line that may have undefined slope is ax + by = c The slope, if it exists, is -a/b; when the slope is undefined, b = 0. If we define two lines this way, ax + by = c dx + ey = f then the product of the slopes will be -1 if (-a/b)(-d/e) = -1 which can be rewritten as ad = -be This form does not require b and e to be non-zero, so it can be used for all lines. The x and y axes are 0x + 1y = 0 1x + 0y = 0 so our condition is 0*1 = -1*0 which is indeed true. And if the first line is horizontal (a = 0), the second line will only be perpendicular to it if e = 0, making it vertical. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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