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#1 2011-10-08 13:18:26

MathsIsFun
Administrator
Registered: 2005-01-21
Posts: 7,535

Vectors

Just finished these pages:

Vectors
Dot Product
Cross Product
Unit Vector

Take your time, read through them, and let me know what you like/don't like and any errors you find or suggestions you have, cheers!


"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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#2 2011-10-08 15:19:29

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 86,676

Re: Vectors

Hi;

That is a lot of stuff! Still looking at the vector page.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#3 2011-10-08 17:36:23

MathsIsFun
Administrator
Registered: 2005-01-21
Posts: 7,535

Re: Vectors

Too right it is, a long time in the making, doing and redoing images etc.

But it has to be as easy to understand as possible while still being correct and covering the topic.


"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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#4 2011-10-08 18:35:55

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,676

Re: Vectors

Hi;

The dot product page looks fine.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#5 2011-10-08 23:03:49

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,525

Re: Vectors

hi MIF

nice pages!!!

i like the jokes on the cross product page but the second one i don't understand very much. smile

anyway,keep on the good work!


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#6 2011-10-08 23:12:08

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,676

Re: Vectors

Hi;

Cross product page is fine. It does contain a lot. I am getting it so anyone can.

I had a hard time with the mosquito.

Unit vector page also looks okay.

Very good work!


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#7 2011-10-08 23:14:42

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,525

Re: Vectors

hi

yes,yes they are.

that's the one i am talking about.


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#8 2011-10-08 23:27:56

ganesh
Moderator
Registered: 2005-06-28
Posts: 14,445

Re: Vectors

Hi MathsIsFun,

The page on vectors is well made, the operations on vectors are simple and easy to understand. Thanks!


Character is who you are when no one is looking.

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#9 2011-10-09 08:43:58

MathsIsFun
Administrator
Registered: 2005-01-21
Posts: 7,535

Re: Vectors

Thanks guys!

Note: A mosquito can be a vector of disease. A mountain climber scales mountains. Maybe it is better off without that joke?


"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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#10 2011-10-09 12:11:51

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,676

Re: Vectors

Hi MIF;

The joke is fine.

I have not given it too much thought but I was wondering about the 59.5 degree problem. The having to round sort of makes it look like something unusual is happening here. Can you get an integer angle in there instead?


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#11 2011-10-10 09:15:38

MathsIsFun
Administrator
Registered: 2005-01-21
Posts: 7,535

Re: Vectors

bobbym wrote:

I have not given it too much thought but I was wondering about the 59.5 degree problem. The having to round sort of makes it look like something unusual is happening here. Can you get an integer angle in there instead?

If the angle is 60 degrees then the first a.b gives 65 instead of 66, so that won't help.

I could play with some other lengths and angles and may get lucky ....


"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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#12 2011-10-10 14:33:20

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,676

Re: Vectors

Hi MathsIsFun;

I have been playing with it and have not been lucky.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#13 2011-10-11 11:07:38

MathsIsFun
Administrator
Registered: 2005-01-21
Posts: 7,535

Re: Vectors

I think it would be nice to also write about vectors vs matrices.

Can we be so bold as to say a vector is a matrix with only one row (or one column) ... ?

Then show the dot product of a vector vs matrix multiplication (a series of dot products)

Then what else?


"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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#14 2011-10-11 12:50:58

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,676

Re: Vectors

Hi MIF;

Can we be so bold as to say a vector is a matrix with only one row (or one column) ... ?

Yes.

Then what else?

Eigenvalues! Markov Chains!


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#15 2011-10-11 19:18:47

bob bundy
Moderator
Registered: 2010-06-20
Posts: 6,393

Re: Vectors

hi MathsIsFun,

These pages are definitely up to your usual high standard.

I think the jokes could stay but you should perhaps title them as jokes.  Many of your readers do not have English as their first language and also have cultural differences.  You might confuse them.  roflol

I'm sure a vector is a one dimensional matrix.  I would prefer as a column.  Then you are set up for matrix transformations of a vector.

In general, your on-line teach yourself pages are invaluable and I often refer 'posters' to them.

But the best, by a long way, in my opinion, are the ones that are interactive or animated in some way.  eg.  the Quidcunx is absolutely brilliant.  It shows something you just cannot get from a text book or even from an experiment.  And the graphical plotter is just so useful; you'll find my help posts are full of screen shots using it.

And the sine graph on the trig index page is such a good introduction to the topic(s).

I realise such features must take you a long time (much longer than the 'static' pages) but with them you are offering the viewer something they cannot get from any other source.  And that makes them priceless!  smile:):)

goldstar.GIF

Bob

Last edited by bob bundy (2011-10-11 19:22:26)


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#16 2011-10-11 22:26:06

MathsIsFun
Administrator
Registered: 2005-01-21
Posts: 7,535

Re: Vectors

Thanks Bob!

The interactives are time consuming, but I do enjoy making them. If you have any suggestions for ones you would like, just let me know.


"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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#17 2011-10-11 23:13:46

bob bundy
Moderator
Registered: 2010-06-20
Posts: 6,393

Re: Vectors

hi MathsIsFun

This is the one I've dreamed about ever since the first computers crept into the classroom.

Aim:  To improve understanding of the decimal numbering system.  To show that the number line is infinitely subdividable and that it has similar properties at every level of zoom.  To help students to understand where, say, 1.57 lies in relation to 1.5 and 1.6.  To show that 1.2 and 1.20 are at the same place on the number line. ....... I'm sure there would be loads more uses once it exists.

Objective:  To show visually that the number line extends indefinitely in both directions, and by zooming in that the space between two consecutive units is subdivided into tenths, and then by zooming again that the space beween two consecutive tenths is subdivided into hundredths and so on.

What you would see:  Initially the viewer is presented with a number line from say ... -4 to +6 ....  A scale shows divisions at each unit.

The viewer can choose to scan to the right or to the left.  Numbers drop off at one end as new numbers appear at the other.

The viewer can choose to zoom in or out.  Let's say the middle point of view is 1.  By zooming in by x10, the line now shows ... 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, 1.5 ....

Now let's say the viewer moves right so that 1.2 is in the centre of view.

By zooming in x10 the line now shows ... 1.15, 1.16, 1.17, 1.18, 1.19, 1.20, 1.21, 1.22, 1.23, 1.24, 1.25 ....

The image below shows my feeble attempt at this sort of thing using a fixed screen shot from Sketchpad.  In my dream version you can scan along and in and out to any (reasonable) degree.

OK. Well you did ask.  If you can do that I believe it will make a major contribution to the understanding of how decimals 'work'.

Bob  smile

View Image: MIF decimals.GIF

You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#18 2011-10-12 08:39:22

MathsIsFun
Administrator
Registered: 2005-01-21
Posts: 7,535

Re: Vectors

Like an enhancement to this: Scrolling Number Line

How do you envisage zooming? With "Zoom x2", "Zoom x10" buttons?


"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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#19 2011-10-12 08:49:24

bob bundy
Moderator
Registered: 2010-06-20
Posts: 6,393

Re: Vectors

hi MathsIsFun,

I hadn't discovered that one.  Like it!

Yes, extending to allow zooming would be great!

I suppose zooming by x10 would be best for the decimal system.  But I'd like a bit of the previous level to be 'carried across' so the viewer could see where they had come from.

Just want to check something on that page so I'll post this far and come back to it.

OK I'm back.

The default starting position could be -10 to +10.  But I like the click for negative or positive option.

Any chance that 'view larger' could fill the width?  I guess it depends on the viewer's resolution.

(1) Slower scroll option so you can 'home in' more easily on any particular number in the centre.  EDIT Just discovered you can control the speed by moving along the arrow.  That's neat.

(2) Then zoom x10 visibly stretches out from the middle with the tenths coming up as this happens so you see that the unit gaps have grown and been subdivided into tenths.  That's the key step.  I want learners to be able to see the number line at steadily increasing magnification so they grasp the concept that each power of ten can be subdivided into ten smaller divisions, which can be subdivided into ten smaller divisions, which can be subdivided .............

Then back to step (1) for another choice of centre number; the (2) zoom then ....

eg. 

Start at -10 to +10.

Scroll to -8 to +12, so 2 is centred.

Zoom x10 to show from 1.0 to 3.0 with 2.0 centred.

Scroll to 1.3 to 3.3, so 2.3 is centred.

Zoom x 10 to show 2.20 to 2.40 with 2.30 centred.

Scroll to 2.15 to 2.35, so 2.25 is centred.

Zoom x10 to show 2.240 to 2.260 with 2.250 centred.

.......................


Trailing zeros should not be supressed.

As I try this out I can see one limitation developing.  The number of displayed digits is obviously getting larger so there will come a moment when there won't be space to show them all.  It won't look pretty if numbers overlap so I wonder what's the maximum number of digits?

I don't know enough about how you can make the display work properly on everyone's computer whatever the make and resolution.  Hopefully, you do. 

I've done a screen shot of my working to get this far.

I suppose later there should be a zoom out option too.  Same priciple in reverse.

Thanks.  Many, many thanks!

Bob

View Image: decimal number line.GIF

Last edited by bob bundy (2011-10-12 09:38:47)


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#20 2011-10-12 09:17:23

Sarah Rebekah
Member
Registered: 2011-09-25
Posts: 540

Re: Vectors

Thats a great Addtion to the forum cool


Love is the key to life

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#21 2011-10-12 10:26:31

MathsIsFun
Administrator
Registered: 2005-01-21
Posts: 7,535

Re: Vectors

bob bundy wrote:

(2) Then zoom x10 visibly stretches out from the middle with the tenths coming up as this happens so you see that the unit gaps have grown and been subdivided into tenths.  That's the key step.

Probably the most difficult part. So it takes 1 or 2 seconds to perform the zoom. And part way through (when gaps are big enough) the line gets further sub-divided by ten. Is that the idea?


"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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#22 2011-10-12 19:28:16

bob bundy
Moderator
Registered: 2010-06-20
Posts: 6,393

Re: Vectors

hi MathsIsFun,

That sounds exactly right.  As the line is expanding the extreme numbers will gradually be lost off the edges and the gaps will be getting bigger.  So maybe subdivisions can appear at 1 second, and the number labels can jump in at 2 seconds when the expansion is complete.  Yes, I think I can appreciate why this is the hardest bit;  I'm hoping for a (apparently) continuous change from state A to state B.  In practice, I guess you'd make small step jumps that look continuous when seen at speed.

On Scrolling Number Line, if I set it to 'view larger' and place my mouse pointer on the very edge of the scroller arrow, I can see the lines jump a small distance followed by the numbers.  But a tiny amount faster and it seems like a continuous movement. So is it possible to expand outwards in a similar fashion?  Sorry I'm being so demanding, but you did say you enjoyed doing them.  smile

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#23 2011-10-12 19:34:38

MathsIsFun
Administrator
Registered: 2005-01-21
Posts: 7,535

Re: Vectors

That all sounds reasonable, but I will let you know what happens in practice as I try to do it.

I wrote the existing "Scrolling Number Line" using SwishMax, but I now use pure Actionscript, so I will need to do a lot of coding before I even get the basics working. In other words ... it may take some time (days/weeks), but I will keep you posted.


"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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#24 2011-10-12 19:50:46

bob bundy
Moderator
Registered: 2010-06-20
Posts: 6,393

Re: Vectors

Thank you.

This is the one I've dreamed about ever since the first computers crept into the classroom.

Weeks?  That's nothing!  I can wait.  Andrew Wiles took 7 years proving Fermat's Last Thoerem.

Best Wishes!

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#25 2011-10-19 21:12:48

MathsIsFun
Administrator
Registered: 2005-01-21
Posts: 7,535

Re: Vectors

I have made a little progress, bob. May have something to show you in a day or two.


"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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