Wednesday, October 24, 2012

What is Perpendicularity?

"Perpendicular" is the term used in mathematics to describe two lines that intersect at right angles.  I recently introduced this concept in a separate post about the definition of perpendicular lines, but I thought it might be interesting to go into a little bit more detail about this rudimentary and familiar math concept.  For students who are just learning about graphing lines for the first time, this is undoubtedly sufficient, but for those more familiar with the concept, this post might provide additional insight.

I opened above by stating that we are talking about two intersecting lines that form a 90 degree angle with each other, though this definition can and should be expanded.   Though technically correct, there is more to the concept of "perpendicularity," as it's called when talking about this subject.  More correctly, this term applies to not only lines (and line segments), but planes (surfaces, not airplanes!) as well.  At first this may sound advanced, but if you think about it, it is completely obvious. If you stack two books together in a perpendicular arrangement, you essentially are viewing the intersection of two planes.  Lines are drawn on paper, but planes are like the three-dimensional versions that have depth., and you can readily find examples in real life of things that are at 90 degrees to each other.

That leads me to a second point about perpendicular lines and planes.  I have said that they intersect at 90 degrees, which is true again, but more accurately and mathematically you can say that the lines form two "congruent adjacent angles" (Wikipedia link).  This means that if you look at the perpendicular intersection in a T-shape, you see two angles next to each other (adjacent) that are the same (congruent).  And by the rules of geometry, since the angles along a straight line must add up to 180 degrees, this obviously means that the intersection must be composed of 90 degree angles.  Furthermore, if the lines extend through each other, the rules of geometry state that all angles around a point must sum up to 360 degrees, so again we have 90 degrees for each - four right angles.

Perpendicular lines

Perpendicular planes

Here's some additional information that you might find interesting.  The term "perpendicular" itself can be an adjective describing the lines, as in "the perpendicular lines are written in red ink."   Alternatively, it can be the noun, as in "the perpendicular to the ground rises to the sky."   Sorry, those examples aren't very creative!  Also, another word for perpendicular is "orthogonal," which can be used the same way to describe right angle intersections.  "Orthogonality"comes from the Greek for "straight angle" and refers to lines and planes at ninety degree angles, much like "perpendicularity."

I know these math definitions probably aren't going to make solving your graphing problems any easier, but I thought that it was good information to know!  It's a very important but basic concept that is introduced very early in math education, so hopefully I have explained it simply enough even for beginners to grasp.  Please click the +1 button below to share my post, and you can even tweet about my site if you like it!

Sunday, October 21, 2012

Definition of Perpendicular Lines

A lot of traffic coming to my site lately has been specifically seeking to learn more about perpendicular lines.  At this point in the new school year, many math classes are just beginning to study graphing, and so perpendicular and parallel lines are often discussed along with other basic graphing concepts, such as slopes and intercepts.  In this post, I would like to give you a definition of perpendicular lines, maybe include a bit of review of some other graphing notes, and go over a few example questions involving them that you are almost guaranteed to encounter in your mathematics courses.  By the end of it, you should be an expert at recognizing and graphing perpendicular lines.

Let's start with the perpendicular lines definition.  It specifically is talking about the relationship between two different lines who intersect at a 90 degree angle.  Two lines that cross at 90 degrees are said to be perpendicular to each other.  A good example of this is the familiar x-y axis.  The y-axis is perfectly up and down with absolutely no slant to the side, whereas the x-axis is perfectly left to right with absolutely no deviation to the vertical.  You can easily see that they form right angles where they meet.  However, it is important to realize that the lines themselves can go in any direction, not strictly up/down and left/right.  The only critical part is that they intersect at 90 degrees.

On a side note, a corollary to this definition is that the four angles created by the intersection of two perpendicular lines are all 90 degrees.  Comparatively, the intersection of two non-perpendicular lines results in the formation of two acute angles (less than 90 degrees) and two obtuse angles (more than 90).  This also demonstrates the geometric law which states that the sum of the angles around a point equals 360 degrees.  This is a useful rule to remember when solving geometric proofs.

So then, now that you understand what this word means and how to visually recognize it on a graph, you may then wish to prove that your two lines do indeed meet the criteria.  How would you even go about that?  How can you determine if two lines are perpendicular?  Well, to do this, you need to know the mathematical equations of the lines… or, more specifically, you need to know the slopes of the two lines.  (Recall that the slope of a line is most simply expressed as "rise over run", which represents the ratio of vertical change to horizontal change.  The slope of the line, often abbreviated by "m", is easily solved by comparing two ordered pairs, and then performing the slope calculation m = y2-y1 / x2-x1.)  True perpendicular slopes will have the following relationship:

m1 = -1/m2

In words, this means that the slope of the first line is equal to the negative inverse slope of the second line.  Looks a bit complicated, but it's not really.  Let's take a look at an example.

Consider the lines described by the equations y = 2x - 1, and y = (-1/2)x + 2.  Are these perpendicular?

This is an example of perpendicular lines.
If you want to get some practice graphing perpendicular lines, you can go ahead and plot these curves.  They are already expressed in standard form, so it is simple to determine the slopes and y-intercepts, and you can also readily generate a table of values to plot points along the lines.  That's a great way to show that you know how to graph the lines, and in the end, you would end up with two lines crossing at 90 degrees.  But that would take an awful lot of time on a test to find a solution that can be found much more quickly and easily.  Just consider the relationship that I explained, and see if it is true in this example.  You can see that the slope of the first line is m = 2, and the slope of the second line is m = -1/2.  This precisely fits the mathematical description of perpendicular lines.  You don't even technically need to graph it out to be able to answer this!  Of course, a wise plan of attack for solving this problem would be to check this relationship first, and THEN graph it out to show that you are correct.  This comes from personal experience.  Always check your work!  ;)

With this information, you should be able to see that all you need to know about lines are their slopes to be able to say whether they are perpendicular or parallel.  Recall, parallel lines have the exact same slope.  The trick when working with this type of question is to realize the the intercept values can be 2 and 3847234, or absolutely anything else at all.  They equation of the line may look completely and extraordinarily different for each, though the only important part of them is their m values.  Keep this in mind, and don't get intimidated by complex and scary-looking equations!

Another type of question might ask you to determine the equation of a line perpendicular to a given line through a specific point.  This takes a bit more work, but it is based on the same concepts.  Let's try a question like this.

Find the perpendicular line to the line y = 2x - 1 that goes through point (4,0).

Here's the approach I would take to solve this.

  1. First, recognize that you are given one of the lines' equations, so from that you can easily find its slope. 
  2. Second, from the first slope, you can use the perpendicular relationship to determine the slope of the second line.  
  3. Third, since you now have the slope and a point that lies on the second line, you can substitute numbers into y = mx + b to solve for b, and then rewrite in in terms of x and y to give the final equation.  

I will leave the actual work for you to try yourself, but the line in this case is the same as above, y = -1/2x + 2.

A third type of question might ask you to determine the perpendicular bisector for a given line segment.  A bisector is a line that perfectly splits another line into two equal pieces, but it can slice through at any angle.  On the other hand, a perpendicular bisector is one the does this at precisely 90 degrees.  If you can first determine what the exact midpoint of your line segment is, you can then apply the approach that I outline above to solve this question as well.

There is one last important point that I would like to make about this topic, and it is about notation.  You are not incorrect to simply state that "line AB is perpendicular to line CD" (or whatever your lines are called), but the shorthand symbol to show this is an upside-down T shape, ⊥.  The keyboard character is called the "up tack", though this term is more applicable to lattice theory, type theory, and logic.  I believe it is more appropriate to simply call it the "perpendicular sign."  So, in this case, you would simply state your answer as AB⊥CD.  That's it.  It's much simpler!

Finally, I thought I would just throw in a bit of trivia that I came across while researching this topic.  Who knows… you might be able to impress your teacher!  The word "perpendicular" originally arose in the late 14th century, and its etymology shows that it came from the Latin word "perpendiculum", which means "plumb line", and "perpendicularis", which means "vertical, as a plumb line".  A plumb line was a simple device which was composed of a small weight on the end of a string, and when holding it up, gravity pulls the weight straight down and the string represents perfectly vertical.  In relation to the ideally perfectly horizontal ground, you can see how they came up with this term.  It's not overly useful information, but you never know where extra trivia might come in handy.

So, with that information, you should now know lots about this subject, and now have no problems graphing perpendicular lines or analyzing and identifying them in either graphs or equations.  There are several different variations to the questions that you may encounter, but if you understand the basics of what it is that defines two perpendicular lines, then you should have no problems in coming up with the appropriate solutions!  Please let me know in the comments below if you would like any further explanation or examples, and don't forget to +1 my post below and follow me on Twitter!  I'm @MathConcepts.  You can even click here to tweet about my post!  Be sure to visit my follow-up post that discusses a bit more of this concept of perpendicularity.

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