Saturday, May 05, 2007

Pythagoras's Theorem.

It seems that most maths book, no matter what topic they're about, have a mention of Pythagoras's Theorem. I mean obviously they have reason to do so, because most books use the theorem as an example to introduce irrational numbers amongst other things.

Since this is 'technically' speaking a maths blog, for completeness I decided to have a post about Pythagoras's theorem. However there is a catch- I present you Pythagoras's Theorem using vectors! Yes your worst nightmare has come true. (well mine has anyway!)

To begin-the definition of the dot product can be found in a previous post (although you may have to scroll down a bit). So now to some properties of the dot product.

:

  • (symmetry)
  • (linearity)
  • (positive definite)
The length of a vector is defined as: and


Pythagoras' Theorem



Proof


It follows immediately that (i.e. u .v=0).


I have been lazy, and not written on the side when I used which property (eg symmetry). I shall on this occasion blame LaTeX and wish that one day blogger will be LaTeX friendly. I realise that I have foolishly chosen to underline my vectors, however it was too late when I realised that on computers they're generally \textbf{written like this}. Since I wanted to be consistent and because once again I'm lazy, I decided to continue the way I had started. The proof has obviously been taken from one of the books I own, however I'd written it on a post it note a while back, and so can't remember which book. A good question would be why did I write it on a post it note! Well I was so excited that vectors were taking over everything, that I wanted to show Po. Alas, Po has yet to do vectors so I stuck the post it note on my wall. (Until the time arises when it will be needed!)

Now I'm experiencing a certain peaceful feeling. Does Pythagoras's Theorem have that affect on everyone, or is it just me? I mean whatever may happen to my blog from now on, at least I have posted Pythagoras's Theorem!


PS: The powers that be made me- don't shoot the messenger! And erm... go Arsenal!

5 comments:

egm said...

Check out this Latex for blogger plugin.

As for the Arsenal thing, I've seen quite a few ManU folks saying they supported them. For one who doesn't follow football much, would you mind explaining the significance?

beans said...

Thanks, I'll see what I can make of it. :)

My new post will answer all questions! (I wasn't in the right state of mind to post anything yesterday :D)

steve said...

The trouble with that is that it requires Firefox with Greasemonkey. If you wish to use mimetex then all you need to do is type something like <img src="http://www.forkosh.dreamhost.com/mimetex.cgi?c=\sqrt{a^2+b^2}">

The mimeTeX site has the details.

I would like to find a volunteer for a similar public server for LaTeX. Meanwhile I've listed the public sites I know about that produce images and documents here.

egm said...

Steve, I did try out the mimetex and it works (don't know why I didn't think of it before commenting!). I tried installing it on my hosted domain, but ran into problems of the script not executing. As soon as I rectify this with my host, I should be able to volunteer hosting.

beans said...

I realised that greasemonkey is trouble! (Have gotten rid of it).

Thanks for the links. I'll keep them in mind for any future posts using LaTeX. :)