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LostInParadise's avatar

What do you think of this explanation for why perpendicualar lines have negative reciprocal slope?

Asked by LostInParadise (17925 points ) January 4th, 2013

This is something taught in high school math that is usually required to be accepted on faith. Here is my intuitive explanation, using the following diagram (it might be somewhat better as a video presentation).

Start with the point A = (5,2). Imagine the x and y axes as wires soldered together. Rotate the x and y axes so the x axis becomes the x’ axis going through A. The slope of the x’ axis is 5/2.

The y axis rotates to become the y’ axis. Since the y axis rotates through the same angle as the x axis, the angle between the y’ axis and the y axis is the same as the angle between the x and x’ axis. If we move down 5 and over 2, the point B that we get should be on the y’ axis. The slope of the y’ axis is therefore -5/2 and is of course perpendicular to the x’ axis.

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7 Answers

Mariah's avatar

I think it has potential, but I think as of now you’re taking it on faith that the lines are perpendicular. I understand your explanation about fixing them in place, but it is not proven that the x and y axes would lie through A and B when rotated. It wouldn’t be hard to add that bit on, I think.

LostInParadise's avatar

What I was trying to get across is that the lines are just a rotation of the coordinate system. Those are not the words that I chose, because I thought it would be easier to see if you imagined the two axes soldered together, so they would maintain their 90 degree angle and the amount of rotation of one was the same as the amount of rotation of the other. An animation might make it clearer by showing the two axes rotating together.

Mariah's avatar

Nah, I get what you were trying to convey. Maybe I’m being too rigorous. You’re showing it graphically, but it’s not really proven.

It’d be trivial to do so, though. The triangles highlighted in yellow are congruent, with equal angles lines together at the origin. Therefore each axis has been rotated the same number of degrees away from the coordinate system axes, and they maintain their orthogonality.

LostInParadise's avatar

In a more advanced class, an exercise could be given to provide a formal proof. I wanted to show the motivation for the relationship, which is something that is frequently lost in class.

At a more advanced level, you can do a similar demonstration of why the derivative of an inverse function is the negative reciprocal of the derivative of the function. I once saw that done by a teacher in an engineering class. I did not take many engineering classes, because, like you, I tend to get hung up on wanting to see rigorous proofs.

dxs's avatar

It sort of confused me at first, but once you said that the lines are just a rotation of the coordinate system, I got what you were getting at, so you definitely want to say that to introduce what you are talking aout. I think that this is pretty justifiable as a method of explaining why the slopes of perpendicular lines are negative reciprocals. It is too bad that most rules of math in high schools are “accepted on faith”, but I don’t see this as completely necessary to explain since it is a bit complex in my opinion for a high school level (this method at least). Nonetheless a great way of explaining it.
My high school Pre-calculus teacher was terrible (should have been fired in my opinion), and he was just like what you said about “accepting on faith”. He would rattle off postulates and laws and what not from a powerpoint for about ¼ of the class and then assign homework. He never bothered to explain why the rules were the way they were, which really bothered me. I do think that most people hate math and don’t care, but people like me who are really into it like to know why rules are the way they are so that we can go beyond applying rules to situations and use logic to solve problems.

LostInParadise's avatar

I find it interesting that you found it easier to understand when I said rotation of coordinate system. I was afraid that people might not understand that. I do think that an animated video presentation would make it easier to understand.

Maybe if students were shown why things worked early on, they might not hate math so much. I once taught a pre-calculus course at a community college as adjunct faculty. My intention was to give explanations for everything. After the second day, two students spoke to me after class. They said that they appreciated what I was trying to do, but they were just taking the course because it was required. All they wanted to know were what equations to use. They then switched to another pre-calculus class. I guess I was starting too late in their education for explanations.

dxs's avatar

@LostInParadise Definitely. I remember math class always being dissed by everyone who had an impact on me. I even specifically look back at it and remember hating it solely because of that reason since I started to finally like it in middle school. I’m considering teaching it at a secondary level, but I feel like I am too interested in math to do that. I just hate dealing with immaturity and tattletales and nose-picking at the elementary school level.

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