Vector Physics: Proving Pythagoras' Theorem

[Cheman]

Vector Physics...

I can completely understand why the idea of vectors works for distances. ie - you can use pythagora and trig to work things out. However, why should that be the same when we are dealing with forces, velocity, etc? In other words, why if we represent forces as certain lengths should they obey Pythagoras' theorum? Please could someone proove why this works to me - maths in explanation is wellcome.

Thanks in advance.

 

[Nenad]

Do you know what the pythagorean theorem is?

 

[selfAdjoint]

I can completely understand why the idea of vectors works for distances. ie - you can use pythagora and trig to work things out. However, why should that be the same when we are dealing with forces, velocity, etc? In other words, why if we represent forces as certain lengths should they obey Pythagoras' theorum? Please could someone proove why this works to me - maths in explanation is wellcome.

Thanks in advance.

The first thing is that forces and velocities do obey the triangle law of addition. So then if you have two of them at right angle to each other, their sum will be the hypotenuse of a right triangle with them (or rather their representative vectors) as legs. Enter Pythagoras.

 

[pervect]

I can completely understand why the idea of vectors works for distances. ie - you can use pythagora and trig to work things out. However, why should that be the same when we are dealing with forces, velocity, etc? In other words, why if we represent forces as certain lengths should they obey Pythagoras' theorum? Please could someone proove why this works to me - maths in explanation is wellcome.

Thanks in advance.

It should be easy to see that if distances add like vectors, velocities, being the rate of change of distance with time, must also add like vectors.

To see this just draw the distances an object moves in some time interval delta-t, and observe that the distances are proportional to the velocities.

You can extend the argument to show that momentum must also add like a vector, since velocities do. (Sticking to Newtonian physics for now, since that's what the question is).

We can extend *that* argument to argute that the rate of change of momentum must add like a vector. But that's a force, F=dp/dt.

This may be a bit abstract, but you did say you didn't mind math :-)

 

[Cheman]

Nenad,
Are you trying to patronise me?! Lol. I said i didnt mind maths - of course i know what pythag is.

Pervect,
Thankyou for your explanation - that's the kind I was looking for. And explanation that actually PROOVES through maths why that rule is true ( ie - because they are all inter-related proportionally) rather than just stating it happens. Thanks.

 

[Galileo]

Notice though, that it is an experimental fact that forces add like vectors.
This is called the 'superposition principle'.
There is no a priori reason to expect that it would naturally hold (which is probably why you posted the question in the first place), but experiments showed it does.