Analytical Geometry

Date: 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/