Convexity and Orthogonality at Saddle Points

Today, the Texas Section of the Mathematical Association of America is holding its annual conference. Like many other professional conferences these days, this conference will be held virtually, and so my contribution to the conference is saved on YouTube and is available to the public.

Here’s the abstract of my talk: “At a saddle point (like the middle of a Pringles potato chip), the directions of maximum upward concavity and maximum downward concavity are perpendicular. The usual proof requires a fair amount of linear algebra: eigenvectors of different eigenvalues of a real symmetric matrix, like the Hessian, must be orthogonal. For this reason, the orthogonality of these two directions is not often stated in calculus textbooks, let alone proven, when the Second Partial Derivative Test for identifying local extrema and saddle points is discussed. In this talk, we present an elementary proof of the orthogonality of these two directions that requires only ideas from Calculus III and trigonometry. Not surprisingly, this proof can be connected to the usual proof from linear algebra.”

If you have 12 minutes to spare, here’s the talk.

Engaging students: Dot product

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Haley Higginbotham. Her topic, from Precalculus: computing a dot product.

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A1. What interesting (i.e., uncontrived) word problems using this topic can your students do now?

For the dot product of vectors, there are lots of word problems regarding physics that you could do that students would find more interesting than word problems self-contained in math. For example, you could say that “you are trying to hit your teacher with a water balloon. Your first try had a certain velocity and distance in front of the teacher, and your second try had a certain velocity and distance behind the teacher. In order to hit the teacher, you will need half the angle between the vectors to hit the teacher. Figure out what angle and velocity you would need to hit the teacher with a water balloon.” This could also turn into an activity, where the students get to test their guesses to see if they can get close enough. There would be need to be something they could use to accurately catapult their water balloon, but that’s a different problem entirely.

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B1. How can this topic be used in your students’ future courses in mathematics or science?

The dot product (and vectors in general) can be seen in physics, calculus 3, linear algebra, vector calculus, numerical analysis, and a bunch of other upper level math and science courses. Of course, not all students are going to be taking upper level math and science courses. However, out of the students going into STEM majors, they most assuredly will see the dot product and by seeing how vectors work earlier in their math careers, they will be more comfortable manipulating something they have already seen before. Also, the dot product and vectors are very useful as a tool to use in upper levels of math and in many different applications of engineering and computer science. In the game design, the dot product can be used to help engineer objects movements in the game work more realistically as a single unit and in relation to other objects.

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E1. How can technology be used?

Geogebra is a great site to use since it has a tool https://www.geogebra.org/m/PGHaDjmD that will visually show you how the dot product works. It’s awesome because you get multiple different representations side by side, so that students who understand at different levels can all get something from this visual, interactive program. They can see how changing the position of the vectors changes the dot product and how it relates to the angle between the two vectors. Also, students will most likely be more engaged with this activity than just doing a bunch of examples with no real concept of how all of these pieces relate together which is not good in terms of promoting conceptual understanding. I think you could also use Desmos as an activity builder to make something similar to the above tool if students find the tool confusing to either use or look at.

References:
https://hackernoon.com/applications-of-the-vector-dot-product-for-game-programming-12443ac91f16

 

 

Visualizing Vectors

From the Math Values blog of the Mathematical Association of America:

Anyone who has taught linear algebra knows how easy it is for students to get absorbed in performing matrix computations and memorizing theorems, losing the beauty of the structures in this foundational subject. James Factor and Susan Pustejovsky of Alverno College in Milwaukee, WI, bring back the visual beauty of linear algebra through their NSF-funded project Transforming Linear Algebra Education with GeoGebra Applets.

The applets are freely available in the GeoGebra book Transforming Linear Algebra Education https://www.geogebra.org/m/XnfUWvvp. Each topic is packaged with a video to show how the applets work, the applet, and learning activities.

Read more about it here: https://www.mathvalues.org/masterblog/visualizing-vectors

Left-Hand Rule?

Misleading pictures in math textbooks always send 10,000 volts of electricity down my spine. Thanks to the right-hand rule, the cross product should be pointing down, not up. This comes from the 2007 edition of Glencoe’s “Advanced Mathematical Topics,” a high-school Precalculus book.

For what it’s worth, this is the same line of textbooks that, in a supplementary publication, said that the rational numbers are not countable.

Engaging students: Vectors in two dimensions

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Sarah McCall. Her topic, from Precalculus: vectors in two dimensions.

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What interesting (i.e., uncontrived) word problems using this topic can your students do now?

For such an applicable topic, I believe that it is beneficial to have students see how this might apply to their lives and to real world problems. I selected the following word problems because they are challenging, but I think it is necessary for students to be a little frustrated initially so that they are able to learn well and remember what they’ve learned.

1. A DC-10 jumbo jet maintains an airspeed of 550 mph in a southwesterly direction. The velocity of the jet stream is a constant 80 mph from the west. Find the actual speed and direction of the aircraft.

2. The pilot of an aircraft wishes to head directly east, but is faced with a wind speed of 40 mph from the northwest. If the pilot maintains an airspeed of 250 mph, what compass heading should be maintained? What is the actual speed of the aircraft?

3. A river has a constant current of 3 kph. At what angle to a boat dock should a motorboat, capable of maintaining a constant speed of 20 kph, be headed in order to reach a point directly opposite the dock? If the river is ½ a kilometer wide, how long will it take to cross?

Because these problems are difficult, students would be instructed to work together to complete them. This would alleviate some frustrations and “stuck” feelings by allowing them to ask for help. Ultimately, talking through what they are doing and successfully completing challenging problems will take students to a deeper level of involvement with their own learning.

 

 

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How could you as a teacher create an activity or project that involves your topic?

I believe vectors are fairly easy to teach because there are so many real life applications of vectors. However, it can be difficult to get students initially engaged. For this activity, I would have students work in groups to complete a project inspired by Khan Academy’s videos on vector word problems. Students would split off into groups and watch each of the three videos on Khan Academy that have to do with applications of vectors in two dimensions. Using these videos as an example, students will be instructed to come up with a short presentation or video that teaches other students about vectors in two dimensions using real world applications and examples.

 

 

 

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How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

Immediately when I see vectors, I think of one specific movie quote from my late childhood that I’ll always remember. The villain named Vector from Despicable Me who “commits crimes with both direction AND magnitude” is a fellow math nerd and is therefore one of my favorite Disney villains of all time. So of course, I had to find the clip (linked below) because I think it is absolutely perfect for engaging students in a lesson about vectors as soon as they walk in the door, and it is memorable and educational. I would refer back to this video several times throughout the lesson and in future lessons because it is a catchy way to remember the two components to vectors. This would also be great to kick off a unit on scalars and vectors, because it would get kids laughing and therefore engaged, plus they will always remember the difference between a scalar and a vector (direction AND magnitude!).  

References:

  1. https://www.khanacademy.org/math/precalculus/vectors-precalc/applications-of-vectors/v/vector-component-in-direction
  2. https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=2&cad=rja&uact=8&ved=0ahUKEwj42PaGqojXAhXKSiYKHTvLD8oQFgguMAE&url=http%3A%2F%2Fwww.jessamine.k12.ky.us%2Fuserfiles%2F1038%2FClasses%2F17195%2FVector%2520Word%2520Problems%2520Practice%2520Worksheet%25202.docx&usg=AOvVaw1IHTinEQtGK4Ww1_JkBhHf
  3. https://www.youtube.com/watch?v=bOIe0DIMbI8

Engaging students: Dot product

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Trent Pope. His topic, from Precalculus: computing a dot product.

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What interesting (i.e., uncontrived) word problems using this topic can your students do now? (You may find resources such as http://www.spacemath.nasa.gov to be very helpful in this regard; feel free to suggest others.)

This website gives an example of a word problem that students could solve, and it has real-world applications. It is not a complete worksheet for students to work on. The teacher would have to create more word problems incorporating the idea of this website. The example on this web page is that you are a local store owner and are selling beef, chicken, and vegetable pies 3 days a week. The owner has a list of how many pies he sells a day and how much they cost. The cost of beef pies are $3, chicken pies are $4, and vegetable pies are $2. On Monday he sells 13 beef, 8 chicken, and 6 vegetable pies. Tuesday he sells 9 beef, 7 chicken, and 4 vegetable pies. Finally, on Thursday the owner sold 15 beef, 6 chicken, and 3 vegetable pies. Now, let’s think about how we can solve for the total number of sales for Monday. First, we would solve for the sales of the beef pies by multiplying the price of the pie and the number we sold. Then we would do the same for chicken and vegetable pies. After finding the sales of the three pies, we would add up sales to get the total amount for the day. In this case, we would get $83 of sales on Monday. The students would do the same thing for the other days the store is open. This is an example of the dot product of matrices in a word problem.

https://www.mathsisfun.com/algebra/matrix-multiplying.html

 

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How could you as a teacher create an activity or project that involves your topic?

An idea I was able to see in an actual classroom during observation this week was the use of Fantasy Football in matrices. A teacher at Lake Dallas High School has her classes in a Football Fantasy League competing against each other. The way they started this activity is that the students have to keep up with the points that their teams are earning. They are doing this by the information the teacher gives them about how to score their players. Each class chooses one quarterback, running back, wide receiver, kicker, and defense to represent their team. The point system is the same as in the online fantasy. For instance, Aaron Rodgers, quarterback for the Green Bay Packers, throws for 300 yards, two touchdowns, and one interception. The points Rodgers earns you for the week comes from taking the several yards and multiplying by the points earned for each yard. Then, do the same for touchdowns and interceptions. After computing this, you will then add the numbers up to get the total points you receive from Aaron for the week. This is using dot product because we have two matrices, which are the stats that the player receives in the game, and the points you get for those same stats. By doing this activity, the students would be working on this aspect of pre-calculus for the entire football season.

 

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How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic? Note: It’s not enough to say “such-and-such is a great website”; you need to explain in some detail why it’s a great website.

Graphing calculators would be a great way to use technology to teach this topic. When computing the dot product of two matrices, there are two ways to do it. One is by hand and the other is a calculator. As the teacher, it would be more efficient for you to see how students are learning the material by having them compute it by hand, but no student wants to do that with every problem. A way the teacher could incorporate solving for the dot product using a calculator in an engaging way would be to have students complete a scavenger hunt. In the scavenger hunt, students will have to solve problems of the dot product to get the next clue and move on to the next. The idea of this would be for the students to show that they can work the calculator and actually get answers. You could have anywhere from five to ten questions for them to solve and decoy answers throughout the room with little mishaps. This would get the students up and moving for this activity

 

My Favorite One-Liners: Part 100

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

Today’s quip is one that I’ll use surprisingly often:

If you ever meet a mathematician at a bar, ask him or her, “What is your favorite application of the Cauchy-Schwartz inequality?”

The point is that the Cauchy-Schwartz inequality arises surprisingly often in the undergraduate mathematics curriculum, and so I make a point to highlight it when I use it. For example, off the top of my head:

1. In trigonometry, the Cauchy-Schwartz inequality states that

|{\bf u} \cdot {\bf v}| \le \; \parallel \!\! {\bf u} \!\! \parallel \cdot \parallel \!\! {\bf v} \!\! \parallel

for all vectors {\bf u} and {\bf v}. Consequently,

-1 \le \displaystyle \frac{ {\bf u} \cdot {\bf v} } {\parallel \!\! {\bf u} \!\! \parallel \cdot \parallel \!\! {\bf v} \!\! \parallel} \le 1,

which means that the angle

\theta = \cos^{-1} \left( \displaystyle \frac{ {\bf u} \cdot {\bf v} } {\parallel \!\! {\bf u} \!\! \parallel \cdot \parallel \!\! {\bf v} \!\! \parallel} \right)

is defined. This is the measure of the angle between the two vectors {\bf u} and {\bf v}.

2. In probability and statistics, the standard deviation of a random variable X is defined as

\hbox{SD}(X) = \sqrt{E(X^2) - [E(X)]^2}.

The Cauchy-Schwartz inequality assures that the quantity under the square root is nonnegative, so that the standard deviation is actually defined. Also, the Cauchy-Schwartz inequality can be used to show that \hbox{SD}(X) = 0 implies that X is a constant almost surely.

3. Also in probability and statistics, the correlation between two random variables X and Y must satisfy

-1 \le \hbox{Corr}(X,Y) \le 1.

Furthermore, if \hbox{Corr}(X,Y)=1, then Y= aX +b for some constants a and b, where a > 0. On the other hand, if \hbox{Corr}(X,Y)=-1, if \hbox{Corr}(X,Y)=1, then Y= aX +b for some constants a and b, where a < 0.

Since I’m a mathematician, I guess my favorite application of the Cauchy-Schwartz inequality appears in my first professional article, where the inequality was used to confirm some new bounds that I derived with my graduate adviser.

Engaging students: Using vectors in two dimensions

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Zacquiri Rutledge. His topic, from Precalculus: using vectors in two dimensions.

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Add vector A and vector B, what do you get? How about when we take the dot product of A and B? What is the magnitude of A, B, and A+B? All of these are basic questions that a teacher might ask their students during a basic high school pre-calculus class. However, how does the teacher respond when a student asks “Where am I ever going to see this again”? In mathematics, a student might never see vectors again unless they take higher math such as Calculus I through III, or possibly Linear Algebra. During the first two courses of Calculus students will continue to expand on the ideas of two dimensional vectors by talking about the path an object might take through the air after leaving a cannon or being thrown off a cliff. Calculus III (or vector calculus) however is a much stronger example of how vectors will be used in further education of mathematics. During this class students will not only look at two dimensional vectors and review simpler ideas, but they will expand these ideas into the three dimensional world creating three dimensional vectors. Here students will discuss what kind of shape or planes a combination of three vectors might create.

A scientific use for two dimensional vectors is in physics. During a physics class, students talk about forces that act on objects as they move or when an object hits another. To do this, students draw vectors to represent the magnitude of the force that is acting on the object and the direction the force pushes or pulls the object. For example, in the previous paragraph it was mentioned about an object being shot from a cannon and the students measuring the path the object might take. In physics, the students might do the exact same thing, but by looking more in depth at the forces acting on the object. Forces might include the force of the cannon firing the object at a certain angle into the air, gravity pulling that object toward the ground, and even the friction of the air on the object as it soars through the air. Each one of these forces is acting on the object as it moves, either helping the object move farther and faster or attempting to slow it down. However, two dimensional physics is not the end of vectors; just like calculus, physics goes on to discuss what happens to objects in a three dimensional world and the forces that act on them. So a very easy answer to give the student asking where he/she will see vectors again is in every day real life.

 

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You have explained to your students that vectors are in everyday life and they still do not believe you. You have shown them countless examples on the board, drawing pictures of airplanes and the paths they fly through the air, objects being dropped from a cliff, objects being shot from a cannon, et cetera, and they still do not believe that vectors have any importance or use! Then you simply ask, has anyone ever seen a show called MythBusters on the Discovery Channel? Now MythBusters is a very well-known show, not only because it has been around for twelve years, but also for some of the crazy things that they test in the name of science. For example, some of my personal favorites include them making a boat out of pykrete, the many episodes on the uses of duct tape, and testing if a bullet dropped at the same time as a bullet shot from a gun will hit the ground at the same time. The great thing about this show is it is full of great examples of how physics affects things in real life. Also, not only do they test the myths, they explain how they are testing them, why they are testing them the way they are, and why it makes sense scientifically or does not. For example, during the bullet episode, they explain that once the bullet is shot from the gun, the only forces acting on the bullet are gravity and air friction. The only forces that would be acting on the bullet dropped would gravity and air friction as well. So in theory these two bullets should hit the ground at the exact same time if they are projected from the same height. By the end of the episode they had proven this by figuring out the best way to set up a live test and using a high speed camera to measure the time it took for each to hit the ground. For a high school class, this would be very easy to draw on a chalk board and walk the students through the thought process of why this happens using vectors to draw out the forces.

 

 

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Finally, the Internet gives us access to a lot of videos. This would allow a teacher who is talking about Mythbusters and their amazing examples of vectors in motion the chance to display quick clips of some of their tests. Of course the teacher will need to have researched a few before class in order to make sure they can be used as vector examples, but after a video has been played the teacher could ask the students to explain why the test was either plausible or false. On a small scale this video, https://www.youtube.com/watch?v=BLuI118nhzc , works great to show how a truck moving at the same speed as a soccer ball being shot from the back cancels the two forces, leaving gravity as the only force acting on the ball. Using vectors, a teacher could explain how one vector is positive and the other is negative of the same magnitude, cancelling the other out. Then show how only one vector on the ball remains, pulling the ball in that direction.

Inverse Functions: Arctangent and Angle Between Two Lines (Part 25)

The smallest angle between the non-perpendicular lines y = m_1 x + b_1 and y = m_1 x + b_2 can be found using the formula

\theta = \displaystyle \tan^{-1} \left( \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \right).

A generation ago, this formula used to be taught in a typical Precalculus class (or, as it was called back then, analytical geometry). However, I find that analytic geometry has fallen out of favor in modern Precalculus courses.

Why does this formula work? Consider the graphs of y = m_1 x and y = m_1 x + b_1, and let’s measure the angle that the line makes with the positive x-axis.

dotproduct5The lines y = m_1 x + b_1 and y = m_1 x are parallel, and the x-axis is a transversal intersecting these two parallel lines. Therefore, the angles that both lines make with the positive x-axis are congruent. In other words, the + b_1 is entirely superfluous to finding the angle \theta_1. The important thing that matters is the slope of the line, not where the line intersects the y-axis.

The point (1, m_1) lies on the line y = m_1 x, which also passes through the origin. By definition of tangent, \tan \theta_1 can be found by dividing the y- and x-coordinates:

\tan \theta_1 = \displaystyle \frac{m}{1} = m_1.

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We now turn to the problem of finding the angle between two lines. As noted above, the y-intercepts do not matter, and so we only need to find the smallest angle between the lines y = m_1 x and y = m_2 x.

The angle \theta will either be equal to \theta_1 - \theta_2 or \theta_2 - \theta_1, depending on the values of m_1 and m_2. Let’s now compute both \tan (\theta_1 - \theta_2) and \tan (\theta_2 - \theta_1) using the formula for the difference of two angles:

\tan (\theta_1 - \theta_2) = \displaystyle \frac{\tan \theta_1 - \tan \theta_2}{1 + \tan \theta_1 \tan \theta_2}

\tan (\theta_2 - \theta_1) = \displaystyle \frac{\tan \theta_2 - \tan \theta_1}{1 + \tan \theta_2 \tan \theta_1}

Since the smallest angle \theta must lie between 0 and \pi/2, the value of \tan \theta must be positive (or undefined if \theta = \pi/2… for now, we’ll ignore this special case). Therefore, whichever of the above two lines holds, it must be that

\tan \theta = \displaystyle \left| \frac{\tan \theta_1 - \tan \theta_2}{1 + \tan \theta_1 \tan \theta_2} \right|

We now use the fact that m_1 = \tan \theta_1 and m_2 = \tan \theta_2:

\tan \theta = \displaystyle \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|

\theta = \tan^{-1} \left( \displaystyle \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \right)

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The above formula only applies to non-perpendicular lines. However, the perpendicular case may be remembered as almost a special case of the above formula. After all, \tan \theta is undefined at \theta = \pi/2 = 90^\circ, and the right hand side is also undefined if 1 + m_1 m_2 = 0. This matches the theorem that the two lines are perpendicular if and only if m_1 m_2 = -1, or that the slopes of the two lines are negative reciprocals.

Inverse Functions: Arctangent and Angle Between Two Lines (Part 24)

Here’s a straightforward application of arctangent that, a generation ago, used to be taught in a typical Precalculus class (or, as it was called back then, analytical geometry).

Find the smallest angle between the lines y= 3x and y = -x/2.

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This problem is almost equivalent to finding the angle between the vectors \langle 1,3 \rangle and \langle -2,1 \rangle. I use the caveat almost because the angle between two vectors could be between 0 and \pi, while the smallest angle between two lines must lie between 0 and \pi/2.

This smallest angle can be found using the formula

\theta = \displaystyle \tan^{-1} \left( \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \right),

where m_1 and m_2 are the slopes of the two lines. In the present case,

\theta = \tan^{-1} \left( \left| \displaystyle \frac{ 3 - (-1/2) }{1 + (3)(-1/2)} \right| \right)

\theta = \tan^{-1} \left( \left| \displaystyle \frac{7/2}{-1/2} \right| \right)

\theta = \tan^{-1} 7

\theta \approx 81.87^\circ.

Not surprisingly, we obtain the same answer that we obtained a couple of posts ago using arccosine. The following picture makes clear why \tan^{-1} 7 = \cos^{-1} \displaystyle \frac{1}{\sqrt{50}}.

dotproduct4In tomorrow’s post, I’ll explain why the above formula actually works.