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Proof of Perpendicularity


Date: 10/23/1999 at 08:18:50
From: Michael
Subject: Logic

Dear Dr. Math,

Could you please help me with the following question?

Prove that two lines (neither vertical) are perpendicular if and only 
if the product of the gradients is equal to -1.

Your assistance is greatly appreciated.

Thanking you in advance,
Michael


Date: 10/25/1999 at 02:49:49
From: Doctor Floor
Subject: Re: Logic

Hi Michael,

Thanks for your question.

Consider two perpendicular lines, which (to make things easy) pass 
through (0,0).

One of the lines is increasing. We can find the gradient of that 
triangle by attaching a right triangle to the line, and divide the 
vertical length (say A) by the horizontal length (say B), so the 
gradient is A/B.

Here is a picture:

   

The line perpendicular to the increasing line can be found by rotating 
this line by 90 degrees clockwise around (0,0). We can rotate the 
attached triangle together with the line. We find (see picture) that 
the gradient now is -B/A.

And we can conclude that the product of the gradients is

     -B/A * A/B = -1

as desired. 

If you need more help, just write back.

Best regards,
- Doctor Floor, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Coordinate Plane Geometry
High School Equations, Graphs, Translations
High School Euclidean/Plane Geometry
High School Geometry

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