Teaching Vectors Properly (For Everyone) - Page 3

In fact, claiming the speed 45 m/s is a vector would lose you marks at school because a teacher would claim it has no direction, but 45 is an element of a the vector space of real numbers or the field of real numbers, so is also a vector.

On February 05 2012 01:26 sukarestu wrote:
I'm one week through year 11 just started my physics class, had 3 lessons (obviously pretty clueless)
Hearing vectors can be a point.. "TRIPPING OUT MAN!!!".. functions "meh.. fine w/e"

Whut? ok.. let me think here, Vector = compound, 45 = atom
45 exists in Vector therefore is vector?
So everything in the field of real numbers is considered a vector?

In fact, claiming the speed 45 m/s is a vector would lose you marks at school because a teacher would claim it has no direction, but 45 is an element of a the vector space of real numbers or the field of real numbers, so is also a vector.

On February 05 2012 00:10 paralleluniverse wrote:
I came across this video on YouTube and it makes me want to puke.

[B]On February 05 2012 01:22 paralleluniverse wrote:
It's a measure. You need to learn measure theory to understand it.

On February 05 2012 01:01 Anytus wrote:
On the whole, I agree with your assessment that the way vectors are taught at an introductory level is usually 'wrong' in the rigorous mathematical sense, but I am not convinced that your solution would fix the problem.

I am a TA for an introductory physics course at the university level and the #1 problem that I see with students' exams and homework is a lack of understanding of how to manipulate vectors. So, I agree that this is a problem.

Your basis in linear algebra is of course correct and completely rigorous and is how I eventually learned about vector spaces for my advanced mathematics. The problem is that most of the students that I teach are not in a position to understand the linear algebra any more than they understand the pragmatic definitions like the ones used in the video you linked to. The concept of a vector space makes the notion of a vector very abstract and for students who aren't majoring in mathematics; it runs counter to the way they have learned math for their entire lives. Most students (at least in the US) learn mathematics only in small chunks that they need to use and not starting from a rigorous foundation. Although most university students could take the derivative of a polynomial function or the limit of some rational function as it approaches zero, they would have a hard time justifying their responses using the formal definitions of a derivative and a limit(I cant even remember this definition most days.....its something like for every epsilon there exists a delta such that......).

Note that we actually do this all the time in mathematics education. We teach a restricted special case in a sometimes inconsistent way to get students to have some basic intuition about the objects and perform basic tasks, then in higher level classes we generalize the notions. I'll list some examples here:
1) The imaginary number i: why assume that there is only one such number? What happens when a function tries to take i as an input, such as Sin(i*x) or Log(2*i*x)? Most students couldn't answer these question after they learn about i to solve the quadratic formula in high school. You need an entire class on complex analysis for that (and even then they might not cover what happens if you assume there is more than one imaginary number (Quaternions).
2) The Dirac delta 'function': here is an idea which even I don't actually understand. Physicists and engineers use and abuse the delta function every day without ever thinking about the fact that it isn't actually a function at all, it is a functional or distribution. It is normally defined as an object with is zero everywhere except a single point and has total integral 1. This definition is patently incorrect if you consider certain sets of series which converge to the delta function, even though their values do not converge to zero almost everywhere, but for most everyone the definition is good enough. More importantly, it helps gives students the intuition of what happens when you use the delta function. I have lots of nagging questions about the delta function because I don't know the theory of distributions, but I get along okay.

I think the crux of the issue is that teaching students the abstract linear algebra version of vectors does not give them a strong physical intuition about how vectors work in physics, and this is the reason that we don't teach it that way. The whole point of our physics course is to develop intuition, not teach specific skills. Teaching vectors as arrows is a much more physically relevant approach given how we deal with objects like velocity. We need concepts like decomposition along a basis vector (which is hard to do/explain with functions in a Hilbert space) and direction (which is nearly impossible to do in that same case) to understand that physics. It is not so important to us if the math is rigorous. I'll note here that quantum mechanics existed for more than 40 years and was used all over the world before its basis was made mathematically rigorous (the idea of a rigged Hilbert space). Ultimately, we teach the way we do because it is best for physics, and I'll let the math teachers speak for themselves as to why they don't delve more deeply into the idea of a vector space.

EDIT: I wanted to add after reading previous responses that I learned about vectors first in high school in a mathematics class and we definitely discussed both dot and cross products. They are important concepts that we mostly expect students to understand when they step into our physics courses.

On February 05 2012 01:34 paralleluniverse wrote:

A few points:

The geometric interpretation of vectors as arrows falls out quite naturally in the teaching method I suggest: Define a vector as an element of a vector space, show that R^2 is a vector space, it follows that the points in R^2 are vectors, then representation as arrows is obvious.

Clearly, this would require more work on the teachers and students part, but I think the much greater clarity this provides is well worth it.

From the vector space R^n, a geometric intuition of decomposing vectors into basis vectors is also natural (this is university level math, and it's usually done in this correct way anyway). I'm not sure what you point on Hilbert spaces is about. The typical example of a Hilbert space is the space of continuous functions, and the basis vectors being the sin and cos function is graphically obvious if you watch an animation of a Fourier series converging.

I wasn't taught the dot product or cross product in high school, but that just because we when to school in different countries.

On February 05 2012 01:34 paralleluniverse wrote:

A few points:

The geometric interpretation of vectors as arrows falls out quite naturally in the teaching method I suggest: Define a vector as an element of a vector space, show that R^2 is a vector space, it follows that the points in R^2 are vectors, then representation as arrows is obvious.

Clearly, this would require more work on the teachers and students part, but I think the much greater clarity this provides is well worth it.

From the vector space R^n, a geometric intuition of decomposing vectors into basis vectors is also natural (this is university level math, and it's usually done in this correct way anyway). I'm not sure what you point on Hilbert spaces is about. The typical example of a Hilbert space is the space of continuous functions, and the basis vectors being the sin and cos function is graphically obvious if you watch an animation of a Fourier series converging.

I wasn't taught the dot product or cross product in high school, but that just because we when to school in different countries.

On February 05 2012 01:26 sukarestu wrote:
Whut? ok.. let me think here, Vector = compound, 45 = atom
45 exists in Vector therefore is vector?
So everything in the field of real numbers is considered a vector?

On February 05 2012 00:41 Djabanete wrote:
If it's a physics class, then an "arrow" is a very seviceable definition that will let you fit the most physics teaching into your lecture.

On February 05 2012 01:29 Plexa wrote:


Speed has no direction btw. That is why the answer is wrong. If the velocity was 45 m/s then the answer is right. speed = length(velocity), speed is an element of R+ (all real numbers greater or equal to zero) - not a field.

but 45 is an element of a the vector space of real numbers or the field of real numbers, so is also a vector

On February 05 2012 00:36 Excludos wrote:
I don't see any problem teaching the first way to highschool kids. Its the same reason they start off saying you can't square root a number below zero (which you can). Because its complex as hell and highschool students don't need it.