Proof of PerpendicularityDate: 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/ |
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