# Parallelogram Rule, Head to Tail Rule

Joined May 19, 2013
214
The two rules of adding two vectors are defined and are correct because that is what we observe in nature?
Or are they a consequence of something else?

If I remember correctly, I think I read somewhere that Newton used the parallelogram rule because this is what he observed to happen in nature.

When adding two vectors at right angles with each other, I can see why the direction of the resultant is that way.
But its magnitude being that way, is not that obvious. I mean, yes, you apply Pythagora's theorem and see what you see. But is not that obvious to me.

#### WBahn

Joined Mar 31, 2012
26,160
What do you mean, "observe in nature"?

Break a two vectors into their orthogonal components and add the components and then combine the result. The "parallelogram rule" and the "head to tail rule" are merely graphical interpretations of this mathematical construct.

#### MrAl

Joined Jun 17, 2014
7,815
Hi,

I dont believe he was trying to prove any vector rules, he was trying to prove laws governing forces in nature. This was i think by using the velocity of an object acted on by two separate forces to show that one equivalent force could do the same thing, and that could be calculated using a parallelogram (remember Newton looked a lot at motion).

So it wasnt about proving if a vector rule was valid or invalid, it was about trying to prove a vector rule applied to a natural situation involving forces or not. The modern version i think seeks to find out why this works with two forces, but again not to prove any mathematical ideas as they were already well established.

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Joined May 19, 2013
214
What do you mean, "observe in nature"?

Break a two vectors into their orthogonal components and add the components and then combine the result. The "parallelogram rule" and the "head to tail rule" are merely graphical interpretations of this mathematical construct.
Yes. It is easier after a life of seeing vectors breaking into components, to say this and take it for granted.

In Euclid's book, for example, Euclid had some definitions and some postulates and then he showed how to construct a circle. He could have said "I'm not gonna do that because it is obvious. Look, this is a circle drawn with my hand"

I think the mistake in my reasoning is that I haven't separated the mathematical object which is the vector from the reality.
I think that we, mathematically, define the vector first, and then show its decomposition and its addition with other vectors.

And I think how you break it is also defined. We could have instead defined that when you break one vector after two arbitrary directions, you take the perpendicular from the tip of the vector to the directions. And then do your mathematical thing and see what happens. I mean, why not. Why necessarily take parallels to the directions from the tip of the vector?

And then after you defined your vector having magnitude and direction and defined how you decompose it, then you see that you can use this mathematical vector to describe natural processes. And you see that how you define your addition of vectors is important. Nature wants you to define it by taking parallels, and not perpendiculars.

PS: in the orthogonal case, parallels would be the same with perpendiculars. But, in the general case of two arbitrary directions, they are not the same.

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Joined May 19, 2013
214
Hi,

I dont believe he was trying to prove any vector rules, he was trying to prove laws governing forces in nature. This was i think by using the velocity of an object acted on by two separate forces to show that one equivalent force could do the same thing, and that could be calculated using a parallelogram (remember Newton looked a lot at motion).

So it wasnt about proving if a vector rule was valid or invalid, it was about trying to prove a vector rule applied to a natural situation involving forces or not. The modern version i think seeks to find out why this works with two forces, but again not to prove any mathematical ideas.
Yes. I agree. I somehow expressed the same idea in the post above.

#### Kermit2

Joined Feb 5, 2010
4,162
As I recall in vector math the only reason for using the parallelogram rule was to move one vector from its position at the foot of a vector to its head so a closed triangle could be formed.
It had no correlation to what actually happens in nature but was strictly a math tool used for calculations that give a resultant vector answer.
I don't recall needing it for Cartesian geometry problems but used it often on circular diagrams with phase and voltage/current values in motor generator design problems.

#### WBahn

Joined Mar 31, 2012
26,160
Nature wants you to define it by taking parallels, and not perpendiculars.
Huh?

How does "nature" want one definition over the other?

#### MrAl

Joined Jun 17, 2014
7,815
Yes. It is easier after a life of seeing vectors breaking into components, to say this and take it for granted.

In Euclid's book, for example, Euclid had some definitions and some postulates and then he showed how to construct a circle. He could have said "I'm not gonna do that because it is obvious. Look, this is a circle drawn with my hand"

I think the mistake in my reasoning is that I haven't separated the mathematical object which is the vector from the reality.
I think that we, mathematically, define the vector first, and then show its decomposition and its addition with other vectors.

And I think how you break it is also defined. We could have instead defined that when you break one vector after two arbitrary directions, you take the perpendicular from the tip of the vector to the directions. And then do your mathematical thing and see what happens. I mean, why not. Why necessarily take parallels to the directions from the tip of the vector?

And then after you defined your vector having magnitude and direction and defined how you decompose it, then you see that you can use this mathematical vector to describe natural processes. And you see that how you define your addition of vectors is important. Nature wants you to define it by taking parallels, and not perpendiculars.

PS: in the orthogonal case, parallels would be the same with perpendiculars. But, in the general case of two arbitrary directions, they are not the same.

Hi there,

I dont think it is that complicated. We see something in nature, we describe it mathematically, end of story A simple example:
We see two purple oranges to the left, two more to the right. We add 2+2=4 purple oranges.
Now, do we sit back and say, "Because we got the right number of purple oranges addition works", or do we say, "Because we got the right number of purple oranges we can apply addition to purple oranges".
The latter is the right choice, because we knew about addition all along, it was this new situation with the purple oranges that we never encountered before.
So we have first observed Nature, then later drew a conclusion. If you want to say that Nature forced us to do that, i guess that is up to you I dont know if i would want to go that far however, because we may observe something more subtle about the experiment later that may change our conclusion so we have to update what Nature "demanded".

Joined May 19, 2013
214
Hi there,

I dont think it is that complicated. We see something in nature, we describe it mathematically, end of story A simple example:
We see two purple oranges to the left, two more to the right. We add 2+2=4 purple oranges.
Now, do we sit back and say, "Because we got the right number of purple oranges addition works", or do we say, "Because we got the right number of purple oranges we can apply addition to purple oranges".
The latter is the right choice, because we knew about addition all along, it was this new situation with the purple oranges that we never encountered before.
So we have first observed Nature, then later drew a conclusion. If you want to say that Nature forced us to do that, i guess that is up to you I dont know if i would want to go that far however, because we may observe something more subtle about the experiment later that may change our conclusion so we have to update what Nature "demanded".
We actually sometimes say "Because we got the right number of purple oranges addition works".

Sometimes you develop new mathematical tricks and then apply them to develop your model, your theory of something from reality. And thus you make predictions and see that it works. Therefore, the mathematical tricks that you used are correct and you can use them further.

This was the case with the discovery of Neptune in the 19th century. It was mathematically predicted. Then they look at the sky and saw that indeed a planet was where they calculated to be.

Thus, because they got right the position of Neptune, that mathematical theory that they used works in nature.

The same with the oranges. It is not that obvious because addition is so embedded in the reality and we use it so often that we don't quite see it as this mathematical trick.
But there is a difference between counting and adding but we do not see it anymore like that. We only see it when large numbers are involved.
I recommend watching this video.

And listen especially to minute 4:40. That "Why?" question. The same here with asking why the vectors are added like the way are added. (in the end, he refers to something else; this why question regarding vectors, I think, is a perfectly reasonable question)

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Joined May 19, 2013
214
Huh?

How does "nature" want one definition over the other?
It wants one definition over the other in the sense that our imagination could define many mathematical tricks that apply in some real situations but don't apply in some other situations.

Nature wants the force of gravity to fall with an inverse square law. Not r, not r^3. But r^2. We could have said "no, it falls with r^5". But that's not how nature works. We also could have defined the vector as being this tool having magnitude and direction and then defined the sum to be with taking perpendiculars. And then say that this vector could represent velocity. But that is not what nature wants, or works. Nature wants parallels. Maybe in the perpendicular case, we could name that quantity some other name, like perpenvectors, and maybe these could be applied in some other area. Who knows? Maybe they really have some utility. I do not know. But in the case of velocity, nature works so that it wants vectors for velocity and not perpenvectors.

#### MrAl

Joined Jun 17, 2014
7,815
We actually sometimes say "Because we got the right number of purple oranges addition works".

Sometimes you develop new mathematical tricks and then apply them to develop your model, your theory of something from reality. And thus you make predictions and see that it works. Therefore, the mathematical tricks that you used are correct and you can use them further.

This was the case with the discovery of Neptune in the 19th century. It was mathematically predicted. Then they look at the sky and saw that indeed a planet was where they calculated to be.

Thus, because they got right the position of Neptune, that mathematical theory that they used works in nature.

The same with the oranges. It is not that obvious because addition is so embedded in the reality and we use it so often that we don't quite see it as this mathematical trick.
But there is a difference between counting and adding but we do not see it anymore like that. We only see it when large numbers are involved.
I recommend watching this video.

And listen especially to minute 4:40. That "Why?" question. The same here with asking why the vectors are added like the way are added. (in the end, he refers to something else; this why question regarding vectors, I think, is a perfectly reasonable question)

Hello again,

I assure you that you are still not interpreting this correctly.
[We actually sometimes say "Because we got the right number of purple oranges addition works".]
But he was not trying to prove MATH, because math was already known. You would only say that if you were trying to prove math, which you dont have to do because you know it works already. There's nothing new about addition, but there is something new in the form of a 'purple' orange, and we want to know the properties of this new fruit. We dont need to use a new and unknown fruit to prove addition works, but we need addition to understand the properties of the new fruit better.
It doesnt matter what we can 'say', because many things we can say will be a side topic and not part of the main argument...that which you seemed to want to know more about from the beginning.

In the analogy, purple oranges are to addition as forces are to vector operations. We wanted to know more about the new fruit and more about forces, but we already knew about addition and vector operations. It would not make much sense to try to prove something using something else who's properties were totally unknown or questionable as of yet.

If you still dont see this point after that, i cant help you Food for thought: what if the purple oranges were capable of reproducing so that when we put 2 and 2 together we got 8 of them ...so then we disproved the mathematical operation of addition?

Joined May 19, 2013
214
Hello again,

I assure you that you are still not interpreting this correctly.
[We actually sometimes say "Because we got the right number of purple oranges addition works".]
But he was not trying to prove MATH, because math was already known. You would only say that if you were trying to prove math, which you dont have to do because you know it works already. There's nothing new about addition, but there is something new in the form of a 'purple' orange, and we want to know the properties of this new fruit. We dont need to use a new and unknown fruit to prove addition works, but we need addition to understand the properties of the new fruit better.
It doesnt matter what we can 'say', because many things we can say will be a side topic and not part of the main argument...that which you seemed to want to know more about from the beginning.

In the analogy, purple oranges are to addition as forces are to vector operations. We wanted to know more about the new fruit and more about forces, but we already knew about addition and vector operations. It would not make much sense to try to prove something using something else who's properties were totally unknown or questionable as of yet.

If you still dont see this point after that, i cant help you Food for thought: what if the purple oranges were capable of reproducing so that when we put 2 and 2 together we got 8 of them ...so then we disproved the mathematical operation of addition?

You have misunderstood me and you put words in my mouth that I did not say. I did not say anything about proving math. Math gets proven by itself. It does not need an experiment from reality to do this.
So, the mathematical theory behind Neptune's discovery was proven to be correct in the usual sense that we all understand it, i.e. the theory is correct. Not in the sense that some theorem from the theory was correct, that some actual math was correct.

I was talking about using math and see if this usage is correct, not prove it. So using that math behind Neptune's discovery happened to be correct. So the math is good.

The fact remains. A scientific approach to devising new theories is to use current math or maybe to discover new math and then use that math to develop a new theory. Then you make predictions and see if it is correct. That's how Neptune was discovered, the Higgs boson was also predicted, the bending of light was also predicted.

With the vectors it could have been the same. You could have defined the vector in a purely mathematical way and then use it in reality.

And nature did force a lot of things on us. Fourier series came from heat study, for example.

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Joined May 19, 2013
214
So in the analogy with the oranges and the addition, by addition i understood the theory used in counting the oranges. Not the actual math that you prove. Of course, it is already proven.

But maybe it has nothing to do with counting oranges. In this case it would have been incorect. The same with vectors. Maybe the force was a scalar so using vectors to describe them would have been wrong, even though the vectors themselves are correct.

#### BR-549

Joined Sep 22, 2013
4,938
The parallelogram rule only applies if you limit the object’s movement to 2D.

Like a bowling ball or cue ball........without spin.

The two forces applied to these objects must be in the same 2D, for the correct resultant.

But when you apply two forces to a charged or neutral object(gravity only) in 3D, you will NOT get that result.

You get a helical rotating vector. No object can move in a straight path in space. The sun completes one helix every eleven years. Jupiter has very little to do with it. I believe this is what set Jupiter’s orbit, not the other way around.

The radius and rate of this vector is 39 magnitudes greater with charge than neutral matter.

This also means that all movement, has at least two vectors, not one.

Force and energy also have at least two vectors. EM radiation is an exception, because one of the vectors is suspended, allowing it to travel in a straight line, but when absorbed(regains a charge source/sink) the vector returns.

The charge helix is too fast for us to notice without special environments and measurements(radio circuit), and the neutral helix is too slow to notice without the same scrutiny(falling body or orbit).

You can ignore all this and only pay attention to one apparent vector, apply parallelogram rule, build a machine, and print money. As we have for the last one hundred years.

But to understand how all science, is really just one science, you must put math in it’s place.

Math is the iron curtain of fundamental understanding.

#### MrAl

Joined Jun 17, 2014
7,815
So in the analogy with the oranges and the addition, by addition i understood the theory used in counting the oranges. Not the actual math that you prove. Of course, it is already proven.

But maybe it has nothing to do with counting oranges. In this case it would have been incorect. The same with vectors. Maybe the force was a scalar so using vectors to describe them would have been wrong, even though the vectors themselves are correct.
Hello,

Well if you are not going to take anyone else's advice then you should have answered your own question by now I will not argue these mundane points any further.

In any case, good luck with your quest.

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