http://www.mapleprimes.com/maplesoftblog/35265-Why-Slopes-Of-Perpendicular-Lines-Are en-us 2026 Maplesoft, A Division of Waterloo Maple Inc. Maplesoft Document System Tue, 09 Jun 2026 14:17:29 GMT Tue, 09 Jun 2026 14:17:29 GMT The latest comments added to the Blog Entry, Why Slopes of Perpendicular Lines Are Negative Reciprocals http://www.mapleprimes.com/images/mapleprimeswhite.jpg http://www.mapleprimes.com/maplesoftblog/35265-Why-Slopes-Of-Perpendicular-Lines-Are http://www.mapleprimes.com/maplesoftblog/35265-Why-Slopes-Of-Perpendicular-Lines-Are?ref=Feed:MaplePrimes:Why Slopes of Perpendicular Lines Are Negative Reciprocals:Comments#comment89026 <p>What is the purpose of the limits?</p> <p>Maple immediately calculates both</p> <pre>tan(x+Pi/2);<br></pre> <p><img class="math" style="display: block; margin-left: auto; margin-right: auto;" src="http://www.mapleprimes.com/MapleImage.ashx?f=71ce0f790c83ab934db71274c9843b70.gif" alt="-cot(x)"></p> <p>and</p> <pre>simplify(tan(x)*tan(x+Pi/2));<br></pre> <p><img class="math" style="display: block; margin-left: auto; margin-right: auto;" src="http://www.mapleprimes.com/MapleImage.ashx?f=7d0027ea25f16d2bde175cfbe00863c4.gif" alt="-1"></p> <p>A simple way to show that the product of slopes of 2 perpendicular lines is -1, is noticing that the slope of a line parallel to the vector &lt;a, b&gt; is b/a, and the vector &lt;-b, a&gt; is perpendicular to &lt;a, b&gt;,</p> <pre>&lt;a | b&gt;.&lt;-b, a&gt;;</pre> <p><img class="math" style="display: block; margin-left: auto; margin-right: auto;" src="http://www.mapleprimes.com/MapleImage.ashx?f=864b3dc2e87da69167387d5ffa204574.gif" alt="0"></p> <p>so the product of the slopes of perpendicular lines parallel to these vectors is</p> <pre>b/a * a/(-b);</pre> <p><img class="math" style="display: block; margin-left: auto; margin-right: auto;" src="http://www.mapleprimes.com/MapleImage.ashx?f=7d0027ea25f16d2bde175cfbe00863c4.gif" alt="-1"></p> <p>_______________<br> Alec Mihailovs<br> Maplesoft Member</p> The latest comments added to the Blog Entry, Why Slopes of Perpendicular Lines Are Negative Reciprocals 89026 Fri, 04 Jun 2010 21:31:32 Z Alec Mihailovs Alec Mihailovs http://www.mapleprimes.com/maplesoftblog/35265-Why-Slopes-Of-Perpendicular-Lines-Are?ref=Feed:MaplePrimes:Why Slopes of Perpendicular Lines Are Negative Reciprocals:Comments#comment89314 <p>The purpose wasn't to verify that the slopes of&nbsp;perpendicular lines are negative reciprocals, but to investigate how a novice might be led to that discovery, using only elementary tools.</p> <p>Of course, Maple evaluates tan(x+Pi/2) directly. But how can this evaluation be reproduced stepwise? It requires the formula for the tangent of a sum. However, that will result in tan(Pi/2) appearing in both the numerator and denominator. Simple evaluation does not work. Hence, right from the start, the limit becomes the appropriate tool.</p> <p>OK, knowing that a stepwise solution is going to require the notion of a limit, how well does Maple handle the limit? At top level, Maple's limit command clearly gets the right result, but the Limit Methods tutor hangs up on this because of the way it is programmed.</p> <p>All of these issues become relevant in the preparation of a lesson for the intended audience, namely, students first learning that the slopes of perpendicular lines are negative reciprocals. That was the point of the discussion - to see if a "derivation" of the result could be constructed within Maple, and if so, how smoothly would it go.</p> <!--break--> <p>RJL Maplesoft</p> The latest comments added to the Blog Entry, Why Slopes of Perpendicular Lines Are Negative Reciprocals 89314 Wed, 09 Jun 2010 00:21:39 Z rlopez rlopez http://www.mapleprimes.com/maplesoftblog/35265-Why-Slopes-Of-Perpendicular-Lines-Are?ref=Feed:MaplePrimes:Why Slopes of Perpendicular Lines Are Negative Reciprocals:Comments#comment89321 <p>It's easier to use another formula, which doesn't require limits,</p> <p>tan(x+Pi/2)=sin(x+Pi/2)/cos(x+Pi/2)=</p> <p>(sin(x)cos(Pi/2)+cos(x)sin(Pi/2)) / (cos(x)cos(Pi/2)-sin(x)sin(Pi/2)) =</p> <p>-cos(x)/sin(x) = -cot(x)</p> <!--break--> <p>_______________<br> Alec Mihailovs, PhD<br> Maplesoft Member</p> The latest comments added to the Blog Entry, Why Slopes of Perpendicular Lines Are Negative Reciprocals 89321 Wed, 09 Jun 2010 03:15:37 Z Alec Mihailovs Alec Mihailovs http://www.mapleprimes.com/maplesoftblog/35265-Why-Slopes-Of-Perpendicular-Lines-Are?ref=Feed:MaplePrimes:Why Slopes of Perpendicular Lines Are Negative Reciprocals:Comments#comment89346 <p>&nbsp;Yes, that avoids the need for a limit because it's the first step in the derivation of the formula for the tangent of a sum. The next step in that derivation is division through by cos(x)cos(y). Since cos(y)=cos(Pi/2)=0, this division step is the reason why a limit has to be taken if the tangent formula is used.</p> <!--break--> <p>RJL Maplesoft</p> The latest comments added to the Blog Entry, Why Slopes of Perpendicular Lines Are Negative Reciprocals 89346 Wed, 09 Jun 2010 17:30:16 Z rlopez rlopez