Vector Physics: Proving Pythagoras' Theorem

In summary, vectors work for distances because they follow the triangle law of addition, and this can be extended to forces and velocities. This is known as the superposition principle, an experimental fact that has been proven through mathematics.
  • #1
Cheman
235
1
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. :smile:
 
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  • #2
Do you know what the pythagorean theorem is?
 
  • #3
Cheman said:
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. :smile:

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.
 
  • #4
Cheman said:
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. :smile:

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 :-)
 
  • #5
Nenad,
Are you trying to patronise me?! Lol. :wink: 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. :smile: 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.
 
  • #6
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.
 

1. What is vector physics?

Vector physics is a branch of physics that deals with the study of physical quantities known as vectors. Vectors have both magnitude and direction, and they are used to represent various physical quantities such as displacement, velocity, and force.

2. How is Pythagoras' theorem related to vector physics?

Pythagoras' theorem is a fundamental principle in vector physics that describes the relationship between the sides of a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

3. How is Pythagoras' theorem used to prove vector equations?

Pythagoras' theorem is used in vector physics to prove equations involving vectors and their components. It can be applied to vector addition, subtraction, and finding the magnitude of a vector.

4. What are the practical applications of Pythagoras' theorem in vector physics?

Pythagoras' theorem has many practical applications in vector physics, including calculating the displacement and velocity of an object, determining the net force acting on an object, and solving problems in mechanics and kinematics.

5. Is Pythagoras' theorem the only way to prove vector equations?

No, there are other methods to prove vector equations, such as using trigonometric functions and the dot product. However, Pythagoras' theorem is one of the most commonly used and intuitive methods for proving vector equations.

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