Excel Vector Calculator - Be Careful How You Use It

I was asked to show this video to students on how to set up their own vector calculator using Excel. The calculator gives you the ability to find the sum of vectors just by entering each vector's magnitude and direction.

Is it useful? Sure it is, but this is where if you don't know what is going on, you may be using it incorrectly! I have a couple of examples to show that.

After you have set up the calculator using the example shown in the video, enter these:

Vector A: Mag=9.8; Ref. Angle=114

Vector B: Mag=16.5; Ref. Angle= -104

Vector C: Mag=11; Ref. Angle=180

If you have set up the calculator correctly, you will get the resultant vector having a magnitude of 20.2 and a direction of 20.4 degrees.

If you don't know any better and I ask you to sketch out the direction of this vector, you would have drawn an arrow that is pointing in the first quadrant of a Cartesian coordinate system, which would be WRONG! In fact, most of my students would do that. It is a natural and automatic tendency to do so since angles are measured counter clockwise relative to the positive x-axis.

If you do a quick sketch and do a "tip-to-tail" vector addition, you will end up with a vector that is actually pointing in the 3rd quadrant! In fact, the true angular direction for this vector is 200.4 degrees (180 + 20.4, the latter is the angle found from the calculator).

The reason for this is that in calculating the angle, one is dividing the y-component by the x-component. This vector has both components being negative and so the division produces a positive value, producing a positive angle. But this angle given by the calculator, if one were to sketch out the vector, is the angle measured from the NEGATIVE x-axis, not the standard positive x-axis. If one remembers lessons from trigonometry, it is why the value of the tangent of an angle is positive in both the 1st and 3rd quadrant.

So the angle given is "correct" if one knows where it is measured from.

Here's another example to try:

Vector A: Mag=12.7; Ref. Angle=45

Vector B: Mag=19.2; Ref. Angle= -171

Entering this into the calculator, you get the resultant vector having a magnitude of 11.7 and angular direction of -30.9 degrees.

Once again, if you simply go by what you get, the tendency here is to think that the vector is in the 4th quadrant, because a negative angle means that it is an angle measured from the positive x-axis but going clockwise.

This is wrong.

The vector is actually in the 2nd quadrant. A simple sketch to do the vector addition will confirm that. The angle "-30.9 degrees" is actually an angle measured clockwise from the NEGATIVE x-axis. For this vector, the x-component of the resultant is negative, and thus, the ratio of the y-component to the x-component is negative, resulting in a negative value of the tangent and the angle. Once again, from trigonometric lesson, the tangent of an angle is negative in the 2nd and 4th quadrant.

What this all means is that a positive angle value is not unique - the vector could be in the 1st or 3rd quadrant - while a negative angle value is also not unique - the vector could be in the 2nd of 4th quadrant. Either do a quick sketch to do vector addition, or look at the sign of the resulting components.

There are two important lessons here. First is that one must know what the numbers mean. Using them blindly without understanding how they come about is risky and may result in the wrong conclusion. Secondly, for this exercise, there is no substitute for doing a sketch and knowing how vectors add. A simple sketch will provide an important sanity check that your conclusion about the vector direction is not wrong.

While this video and the setting up of the calculator is useful, the producer did not go far enough to demonstrate the possible pitfalls in using it blindly. There should have been examples involving what I had presented to tell the viewers what they should be careful about. I just wonder how many people had used this and interpreted their results incorrectly.

Zz.

Is 1/3 smaller than 1/4?

I'm sorry if this is old new, but I just found out about this recently.

I read a rather amusing account on why A&W 1/3 pounder lost out to McDonald's quarter pounder, even though they were both at the same price.

Confused why A&W's burgers weren't able to compete even though the burgers were priced the same as their competitors, Taubuman brought in a market research firm. 

The firm eventually conducted a focus group to discover the truth: participants were concerned about the price of the burger. "Why should we pay the same amount for a third of a pound of meat as we do for a quarter-pound of meat?" they asked. 

It turns out the majority of participants incorrectly believed one-third of a pound was actually smaller than a quarter of a pound. 

I hate to say it, but this is no longer surprising to me. I look back on my take on the public's understanding and perception of science, technology, and math, and the dismal state seems to have persisted. Nothing has changed. In fact, when I said this back in 2010 .... 

As scientists, we cannot forget this, because it explains the fickleness in the support that we get. That overwhelming support that is there one day can easily go away the next day, and not because of some scientific evidence, but possibly because someone else has better bells and whistles.

... I just never expected it to be illustrated so glaringly during the past few years. Many in the public do not have the ability evaluate the validity of a claim or evidence, and science can easily lose its support because someone else has a more attractive message, even without any valid evidence.

What are the odds that this is the root cause of our debacle today?

Zz.

Roger Penrose - Is Mathematics Invented or Discovered?

 Now that he has just be awarded the 2020 Nobel Prize in Physics .... :)

This is a video of a conversation with Roger Penrose on his opinion of mathematics and its ability to describe our world.

 

Eugene Wigner also had written a rather popular essay on what mathematics is and its "unreasonable effectiveness" in describing our world.

Zz.

Simple, Basic, COVID-19 Math

This is highly elementary for most of you. But I've learned a long time ago that what I consider to be obvious and trivial, is not the case for many members of the public. This is one such case because I've heard this uttered in the media, in print, and among some people.

There is a large increase in the number of positive COVID-19 cases being reported in many states in the US. A lot of people, who shall remain nameless, make the excuse that this is due to the increase in testing, and that it shouldn't be alarming. The more you test, the more you find, they argued.

So I'll illustrate this with simple, basic math.

Let's say you have a population of 1000 people. And let's say that 200 of them has COVID-19. This means that 20% of the population has the virus.

If you randomly test 100 people, you'll get 20 people who is positive.

If you randomly test 200 people, you'll get 40 people who is positive.

If you randomly test 300 people, you'll get 60 people who is positive.

So yes, in terms of absolute numbers, the more people you test, the more number of positive results that you will get. HOWEVER, look at the percentage of positive test. No matter how many people you test, the percentage will still be 20%. The absolute number will increase with increasing number of tests, but the percentage (some people call it positive rate) does not change considerably. This is what a lot of people appear to not fully comprehend.

Of course, in real life, the percentage won't be exact, but if we keep having these tests, it will hover around some value and not change systematically or monotonically over time if there is no change in the number of people infected. So what you need is not the absolute number but also the overall percentage of positive result per day, let's say. That is of more importance if you are trying to see if things are getting better, or worse.

The situation, of course, is more complex than this. But the point in all of this is that you simply can't dismiss the increase in numbers by claiming that it is due to an increase in the number of testing. That is not the whole picture. What if you do the above exercise again and instead, you get 50 out of 200 being positive one week, and 90 out of 300 being positive the week after that? Can you still attribute that to increase in testing only?

Zz.

Are Math Jokes Funny?

Let's dissect!



Zz.

What Is The Big Deal With This Math Problem?

For at least a year now, I've seen this math problem being floated about the various new websites. And I don't understand why it is such a big deal.

The problem involves a simple math problem that many students first learned in an intro algebra class:

Now, any child learning something like this would have to also learn about the SEQUENCE of operations that one has to perform to do this correctly. If you simply start to enter this into your calculator in order that it is written, you'll get the wrong answer.

And of course, you have to know that dividing by 1/3 is equivalent to multiplying by 3.

There are many mnemonic guides that one can use to know which one to perform first. In this case, you first perform the division, thus simplifying the equation into:

9 - 9 + 1 = ?

which will obviously leave you with the answer of 1.

This problem is getting rather a lot of publicity because it claims that a lot of people didn't get the right answer for something that seemingly looks very simple. My response to that is: Yeah, so?

When one learns this rule in school, one is given many similar problems of this type. This is not an unusual problem, and certainly something a lot of people will get wrong if they don't remember what the rule is. This is not surprising.

But why is it getting this much publicity?

Zz.

Don't Ask Siri To Divide 0/0

... unless you want a snarky remark about your personal life from her. You might get this response:

"Imagine that you have zero cookies and you split them evenly among zero friends. How many cookies does each person get? See? It doesn't make sense. And Cookie Monster is sad that there are no cookies, and you are sad that you have no friends."

Yowzah!

So, have you tried any other math questions with Siri and got similar amusing responses? Do share!

Zz.

How Big Is The Sun?

Hey, you get to use some of your high-school geometry and trig to make sense of this video!



Zz.

Use "i,j,k" notation instead of "arrow" representation for vectors in Intro Physics?

That is what the authors of this study have found to be more effective in analyzing students understanding and ability to comprehend vector problems. (The paper is available for free.)

First, we replicated a number of previous findings of student difficulties in the arrow format and discovered several additional difficulties, including the finding that different relative arrow orientations can prompt different solution paths and different kinds of mistakes, which suggests that students need to practice with a variety of relative orientations. Most importantly, we found that average performance in the ijk format was typically excellent and often much better than performance in the arrow format in either the generic or physics contexts.

My question is, is this the result of an inherent conceptional problem in the arrow representation, or simply a matter of correcting some of the ways we teach vectors to students?

Zz.

Oliver Heaviside

This month's issue of Physics Today has a terrific brief biography of Oliver Heaviside. If you've studied physics, mathematical physics, or even electrical engineering, then you would have encountered and used the fruits of his labor.

In physics, there many many of these unsung heroes that do not get the public recognition that they should. It is only through articles such as this, and highlighting them in blogs such as this one, that these figures will at least be known to a few more people that have never heard of them.

Zz

Brief Biography of Emmy Noether

To celebrate her 130th birthday this month, the NY Times has an informative biography of "The Mighty Mathematician You’ve Never Heard Of" - Emmy Noether (read it quickly before it is no longer available for free online).

Of course, to many of us in physics and mathematics, her name is quite familiar, and if you need a concise reason why she is so important in physics, this description of the Noether theorem would settle the case:

What the revolutionary theorem says, in cartoon essence, is the following: Wherever you find some sort of symmetry in nature, some predictability or homogeneity of parts, you’ll find lurking in the background a corresponding conservation — of momentum, electric charge, energy or the like. If a bicycle wheel is radially symmetric, if you can spin it on its axis and it still looks the same in all directions, well, then, that symmetric translation must yield a corresponding conservation. By applying the principles and calculations embodied in Noether’s theorem, you’ll see that it is angular momentum, the Newtonian impulse that keeps bicyclists upright and on the move.

I continue to be amazed at many of these women scientists and mathematicians who, despite all they had to endure during the times that they lived in, were able to persevere and produce such profound body of work. It is amazing enough to produce these amazing work. But considering that these women had so many social obstacles that the men didn't have, one can't help be impressed by what they had accomplished.

Zz.

How Many Slaves Would It Take To Pick Those Oranges?

The 8-year olds at the Beaver Ridge Elementary school somewhere in Georgia had a rather "colorful" math homework assignment. They were asked to solve math problems with very eye-raising context:

"Each tree had 56 oranges. If eight slaves pick them equally, then how much would each slave pick?"

"If Frederick got two beatings per day, how many beatings did he get in one week?"

It is never too young to instill bigotry or violence, I suppose.

Supposedly, the possible reason why the math teachers gave such questions was .....

"In this one, the teachers were trying to do a cross-curricular activity," Gwinnett County school district spokeswoman Sloan Roach said.

 Really! What curricular activity were they trying to cross with math? Child-beating 101?

At best, some people used very poor judgement and poor tastes.

Zz.

Can Insurgent Attacks Be Modelled By A Simple Equation?

University of Miami physicist Neil Johnson and his co-authors seem to think so. They have published an analysis of insurgent attacks in Iraq and Afghanistan, and discovered a pattern to such activities.

Johnson and his research team gathered publicly available data on military fatalities in Afghanistan and Iraq. On a graph, the numbers created a distinct, upward curve.

He says it wasn't just a coincidence; those numbers follow a specific mathematical pattern. In this case, the pattern translates into an equation you can punch into a handheld calculator, says Johnson.

I'm typically skeptical of this type of modeling of human behavior. In this instant, the unpredictability of even the action of one person is enough to screw up the predictions, and that in turn could have a chain of consequences.

Besides, once something like this is published, who is to say that these insurgents would not become aware of their pattern, and try to plan their attacks to NOT follow such pattern?

Zz.

PDE Class Rumor Creates Excitement?

Oooh.... a rumor on a possible creation of a Partial Differential Equation class! This is a damn, fine gossip! :)

OK, I read this news report, and then scratched my head, and then read it again, and then said to myself "OK, what did I miss here?" :)

It seems that the rumor that the Physics Dept. at McGill University in Canada might create a class in Partial Differential Equation is causing an excitement not seen since "... they put a flat screen in the foyer has Rutherford Physics .... " Oh my! That's is an excitement!

“PDEs is one of those things that if you want to do physics, it pretty much puts up a wall if you don’t have it,” says McGill Society of Physics Students VP Academic Nina Kudryashova. “It’s so omnipresent.”

Although it’s been brought up, it is unlikely that PDEs will become a requirement anytime soon. “To even give it rumour status is going a little far” Physics Undergraduate Curriculum Committee Chairman Professor Kenneth Ragan says, “and for current [physics] students lacking PDEs, it’s not fatal.”

Physics professors often include higher-level math, like PDEs, in their curriculum on a need-to-know basis: if a particular tool from a math course which is not required for physics majors is needed, the professor will explain it in class.

Er... Hum. When I was an undergraduate student, I took a class on PDE from the Math dept. There wasn't ANY question on whether it was needed or not, since we ALL know that a physics undergraduate NEEDS to know PDE. Nowadays, many physics dept. have courses in "mathematical physics", in which PDEs are covered. I think teaching it on a "need-to-know" basis is highly inefficient, especially when it is taught during the actual physics class where PDE is needed. You are trying to learn both the physics, and the mathematics, at the same time. I've mentioned in my "So You Want To Be A Physicist" essay why this is not the best way to learn physics (read the chapter of mathematical preparation).

It is interesting that, a "rumor" that most of us don't consider to be anything significant, is creating quite an "excitement" among McGill's physics students. Could it be that they are really indicating that there is a need for such a class? Even if it isn't just a course in PDE alone, a mathematical physics class using a text like Mary Boas' "Mathematical Methods in the Physical Science" could fulfill the same needs.

Zz.

Microsoft Mathematics

Heaven knows I don't need another graphing software or mathematical package. But still, when I came across this, I thought that this might be very useful to students, especially those in high school or early undergraduate years. I know that there's probably several freeware or shareware mathematics packages that one can afford without paying the hundreds (or thousands) of dollars for the high-powered mathematics packages. Still, this might be "free" enough that it is interesting.

It appears that Version 4 of Microsoft Mathematics is available for download. A brief description of it can be found here. I haven't used it before, and before I recommend it to people I know who might need it, does anyone else have a review of this or an earlier version of it? And if you have used it for physics purposes, I would like to hear your opinion even more.

Zz.

Mathematics and Physics

I can't say whether I completely agree with the content of this report, but it is certainly interesting to consider.

This news article reports on a lecture given Dartmouth on the interaction between Physics and Mathematics, its common history, and how they are now different but interconnected. The lecturer has an interesting point on when in history the field of physics and mathematics split.

The fields began to divide in the late 1800s, she said, with individuals coming to identify with one of the two fields.

“The big question is, what happened between the early 1800s and late 1800s?” she said. “Why was there a split into mathematics and physics? The answer: Fourier happened.”

Joseph Fourier is best known for his development of the Fourier series, a method of breaking sounds into components. The method explains, for example, why a violin sounds different from a flute, Singer said. But in addition to the Fourier series, Fourier came up with the theory of heat distribution, a model for how heat travels on a thin, metal plate.

“This [theory] lit up a crisis in the scientific world,” Singer said. “And the resolution of this crisis split natural philosophy.”

Natural philosophers had formerly agreed upon the definition of a function as a formula and a formula as a function, according to Singer. But to support his theories, Fourier used functions, but not formulas — contradicting the commonly held notion that the two were interchangeable, she said.

As a result of this unprecedented disagreement, mathematics and physics diverged from one another. Those who agreed with Fourier became physicists and those who didn’t became mathematicians, she said.

Wow. So we can blame it all on Fourier? Or maybe some would think that we can thank it all on Fourier! :)

Zz.

Female Teachers’ Math Anxiety Affects Girls’ Math Achievement

We have seen many previous reports and studies in which teachers can be a significant factors in how well students understand a material. However, I think this is the first one I've encountered whereby there is a gender-specific effect. A new study concluded that if a female teacher has anxiety of her math skills, this will transfer more likely to female students in the class. What resulted is rather interesting: the female students will acquire the impression that boys are better at math than girls, and will also cause these students to perform poorer in mathematics.

But by the end of the year, the more anxious teachers were about their own math skills, the more likely their female students but not the boys were to agree that "boys are good at math and girls are good at reading."

In addition, the girls who answered that way scored lower on math tests than either the classes' boys or the girls who had not developed a belief in the stereotype, the researchers found.

The study was published in PNAS, and you can have access to the paper at the PNAS website, at least, I think you could since it is an open access article.

Zz.

Math Education Researchers in Demand?

Mathematics education researcher in very high demand??!! Who knew??!!!

I certainly didn't, and this must have been flying under the radar for many of us who are not in such a field. So when I saw the title of the article, my first question was "What exactly is "math education research"?"

Mathematics-education research demands a rare combination of interests and skills, says Alan Schoenfeld, a mathematics education researcher at the University of California, Berkeley. Mathematics is important, of course; but math-education researchers also use social-science tools to study teacher behavior, student reasoning, educational equity, technology, and other topics. "A lot people who are brought up in the mathematics and the sciences tend to think the social stuff is soft and therefore not as intellectually interesting or rigorous," says Schoenfeld, who holds a Ph.D. in mathematics. But "math ed., properly done, is actually more challenging than mathematics, and that's because simple systems sit still and people don't."

And they can't seem to find enough people to do this? Wow!

Zz.

Science Helps Unravel Mystery of 'A Hard Day's Night'

I must have lived a musically-sheltered life, because I never knew that a mystery concerning the first note of the Beatles's "A Hard Day's Night" existed till now.

Guitarists and other musicians for more than 40 years -- without success -- have tried to recreate the sound that opens "A Hard Day's Night." While the note played on George Harrison's 12-string Rickenbacker is the most prominent, the two-second duration of the chord is filled with undertones and frequencies that musicians have been unable to identify or recreate.

Now someone has studied the Fourier transform of the note (why this obvious analysis was not done sooner is a mystery to me).

A research student at Dalhousie University, in Canada, used a mathematical calculation known as Fournier Transform to figure out which notes are on the record.

Jason Brown took up the issue in his thesis.

Brown's "Mathematics, Physics and 'A Hard Day's Night'" details what he learned by analyzing the opening chord in a computer sound editing program.

The "mysterious notes," according to Brown's research, come not from guitar notes played by George Harrison or John Lennon or bass notes from Paul McCartney, but a piano played by Beatles producer George Martin.

Mystery solved!

Zz.

What Modern Mathematics Means to You

I suppose if the general public tend to be persuaded more by style rather than substance, then this style of essay should be more attractive to them rather than just saying "basic, fundamental research is important for its own sake, because you just never know what it will spawn in the future".

That is essentially what this excerpt is saying. It is an excerpt from the book "The Education of T.C. MITS --What Modern Mathematics Means to You" written by Lillian Lieber. It is an "illustration" to justify the esoteric part physics and mathematics, and why they are worthy of support.

There are certainly many good points being made here. And maybe the style points are just exactly what the public might be attracted to to pay attention to the content.

So is it working?

:)

Zz.