# Application of perpendicular lines

Endeavoring to bring some flavor of "real world" application to each topic in my community college precalculus class, I find myself struggling to provide some non-geometric motivation for perpendicular lines.

Applications of parallel lines are a dime-a-dozen, as the idea of rate-of-change is easy for students to discuss when there are units involved. The discussion of the relevant units makes the conversation meaningful.

Now, is there some level-appropriate, non-geometric application of perpendicularity? I am having trouble thinking of something that doesn't strictly rely on the angle between the functions ("find the equations for the sides of this square"), tangent lines (e.g. finding the gradient), or a special case with the units ignored ("this thing goes up by 1 each year, and the other thing goes down by 1 each year").

Do you have a go-to example of a non-geometric application of perpendicularity for precalculus students? In your example, do the units tell something useful? If so, I'd love to hear about it.

Incidentally, here's what I'm thinking of as an easy-to-build application of parallel functions for my students:

• Two people begin working at the same job, and each year they will both see a 2000 dollar-per-year raise. Person A comes with experience, beginning at 55000 dollars per year, and person B starts at 45000 dollars per year.
• What do you mean by a "non-geometric application of perpendicularity"? To me that sort of sounds like a "non-arithmetical application of addition" or "non-statistical application of variance". Sep 19, 2017 at 19:43
• John Coleman -- To me, there is nothing inherently geometric about the example I gave about salaries increasing, unless I graph the relevant functions and look at the lines. If two unit-possessing relationships are parallel, then there is often a nice way to describe what makes them parallel them in words, without mentioning points, lines, planes or angles. However, I'm having a hard time providing an example of two relationships which are perpendicular, where the perpendicularity can be explained without mentioning geometric objects (points, lines, planes or angles). Does this make sense? Sep 19, 2017 at 21:06
• It is an interesting question. "Orthogonal" is sometimes used when the geometric meaning isn't so intuitive (e.g. two vectors which dot to zero in some high-dimensional space). Perhaps you could search for simple examples of orthogonality. Sep 19, 2017 at 21:20
– AnoE
Sep 20, 2017 at 18:01
• I don't find where I read a really convincing argument against trying to cook up real-world example when teaching math, but this one comes close: byrdseed.com/beware-real-world We do need to engage students, but real world example are overrated and usually are cheats. Realistic applications are often too difficult to introduce a notion (they need the student to already master the notion), while one can have great fun with fantastic, weird, pop-cultural examples. Sep 21, 2017 at 20:00

There's a reason why you can't find a good non-geometric example: when the dimensions of the axes on a graph are distinct, perpendicularity is units-dependent. There is thus no natural aspect ratio with which to draw the graph. If you change the scale for just one axis, you destroy any perpendicularity that was present, thus demonstrating that it was an accident.

• Scale does not affect perpendicularity. The scale can be anything, as long as it's the same for both axes. The same goes with "scale" replaced with "units". Sep 20, 2017 at 8:53
• @JoonasIlmavirta Davis explicitly says "when the dimensions of the axes on a graph are distinct" in which case there is no principled reason not to change the scale of just one axis. OP's example of parallel involves dollars and years. There is no reason at all to have e.g. 1 dollar and 1 year represented by the same distance on an axis. The scale choice that you make wouldn't effect two lines being parallel -- but it would effect if the lines are perpendicular. This is an insightful answer. Sep 20, 2017 at 10:41
• @JohnColeman I agree, this is a fine answer. I don't fully agree with "perpendicularity is units-dependent". It is true that if different axes have different units, then the concept is meaningless, and this has to do with scaling one axis instead of the other. What I wanted to say is that scaling the whole plane (changing units on both axes) makes no difference for perpendicularity, so one doesn't really need a natural scale. Sep 20, 2017 at 12:08
• @JoonasIlmavirta: I reworded the answer to avoid making the statement overbroad. Sep 20, 2017 at 13:07
• @DavisHerring Looks good! +1. Sep 20, 2017 at 13:53

Given a set of sites determine which points are closer one of the sites than any of the others? If closer is measured by Euclidean distance then given the distinct points A and B the points equidistant from A and B lie on the perpendicular bisector of the segment A and B. The regions one obtains are known as the Voronoi diagram associated with the sites, and pieces of lines perpendicular to the lines joining pairs of sites play an important role. The sites may be post offices or schools so the "cells" of the Voronoi diagram are the postal districts or school districts. Here is a primer for these ideas: http://cs.brown.edu/courses/cs252/misc/resources/lectures/pdf/notes09.pdf

• Great example! For further work with Voronoi diagrams you can note the John Snow maps dealing with the Cholera outbreak in London, or Descartes diagrams for his vortical model of outer space. Sep 26, 2017 at 21:06

Builders use plumb lines and levels to determine vertical and horizontal directions (which are perpendicular) in order to ensure that floors are horizontal and walls are vertical.

When dealing with vectors (in the plane or in space), which are used constantly in statics and dynamics, one often represents each vector as the sum of a pair of perpendicular vectors. Flip through any introductory engineering statics or dynamics book for many such examples.

This is a bit fanciful, and likely too geometric for your purposes, but if your students have encountered trig...

Road intersections are safest if the roads cross at right angles. Suppose the roads are $\sin$ and $\cos$ functions, say $\sin x$ and $\cos(x+\pi)$. Do they cross at right angles? No. Suppose you scale up the second road to the steeper $s \cos(x+\pi)$. What $s>1$ leads to crossing at right angles? Of course to solve this they need the derivative, but perhaps this could motivate slopes.

$\sin x$ and $2 \cos(x+\pi)$ cross orthogonally at $x=-\frac{1}{4}\pi,\, \frac{3}{4}\pi$.

There are many applications of perpendicular lines in high school Physics. My students must often figure out the perpendicular to a given line.

The normal force is perpendicular to the surface on which an object is resting or sliding.

Snell's Law, which governs how a light ray behaves when it crosses the boundary between two media (such as air and glass), requires the line perpendicular to the boundary.

In calculus I class today: Find the point on y=x2 that is closest to (1,0). (Actually, approximate it, using Newton's Method.)

The closest point makes a line segment perpendicular to the tangent line. [True for a point and a line. Is this true for a point and a curve by definition? It is just what came to me in class, when a student asked me to do this problem. I had assigned it but never thought about it carefully. I see that using distance gets me to the same place algebraically, but I haven't yet established the perpendicularity to my satisfaction.]

The student had tried to minimize distance, which is much harder. Using the perpendicular made the problem relatively straightforward.

• Yes, also true for a point $p$ and a curve $C$, not by definition, but because if the min distance were realized by $p x$ with $x \in C$ without perpendicularity, then $x$ could be slid one way or the other to reduce $|p x|$. Sep 22, 2017 at 11:47
• Yes, what I was thinking about. I just didn't feel I had it proved. Ahh, I'm seeing a triangle now. Yes, I like that. Sep 24, 2017 at 2:13

I don't know if you're doing anything with matrices, but I think this would be a nice opportunity to show the difference in approaches to statistical models.

Linear Regression is easily demonstrated using a transpose and matrix multiplication. This model assumes noise in both the $x$ and $y$ variables. An alternative situation involves noise in only one measure, and now you would want to minimize the distance perpendicular to the line of fit rather than the strict vertical distance as in linear regression.

This is Principal Component Analysis, and also can be understood through some basic matrix operations. Of course, there is geometry involved in the 2D case that we can visualize but higher than 3D doesn't quite give way to images.

Here is a site with nice visualizations and a fun example about eating in the UK.

http://setosa.io/ev/principal-component-analysis/

Also, see Wikipedia for a more substantial mathematical presentation.

https://en.wikipedia.org/wiki/Principal_component_analysis