Why are we able to use Components of Vectors?

Question

Last year, I took physics and over the summer I have started to wonder about why many phenomena work the way they do (such as why is $\mathrm{KE} =\frac{1}{2}mv^2$, etc.). I have found answers to all of my questions, except for this one:

Why are we allowed to use trigonometry to get vector components? I understand that if we draw a right triangle, then we can express the adjacent side with $\text{length of hypotenus} \times \cos(\theta)$. But why does this work with velocity and with forces? I have tried to reason through this many times, but I am not able to figure out why we are able to do this. Any ideas?

BTW: My attempts for reasoning through velocity were as follows: If we have a ball moving in two dimensions, and we are given that the ball has $\sqrt{3}$ the speed in the x-direction than the y-direction, then we can conclude that the ball will move in a 30-degree angle. I am not sure how to apply this to forces or acceleration though. For example, why is it that in the case of a ball spinning horizontally while attached to a string, that the y component of the strings tension gets smaller as the centripetal force increases? I can see how we can prove this by using trigonometry, but why does this trig work in the first place?

Answer 1

One way to think of this is that you can take any force. That force can be represented by a vector. You can superimpose any coordinate grid on that vector and decompose it into its x, y, and z components according to that grid. You can think of this as a mathematical trick for designing 3 forces, that when they are act together, create the same effect as the original force.

One of the comments is "Then why does this superposition theorem work?" And the answer to that is that the vectors in the x, y and z direction are completely independent of each other. For example, if you have two particles that collide (in a perfectly elastic collision), not only is the total momentum conserved, but the momentum in each of the 3 axes is also, independently conserved. (That fact came as a real eye-opener to me when I first learned it.) So, after you've done this decomposition, mathematically, you haven't lost anything. The 3 vectors in the x, y, and z direction contain all the information you need to recompute the original vector.

Why are vectors in the 3, mutually perpendicular axes completely independent of each other? That gets harder to answer. It has to do with the definition of spatial dimension: Different dimensions are orthogonal, meaning that forces that act along a single dimension simply cannot affect any of the other, mutually perpendicular dimensions.

And why is that true? That comes down to the question "Why are there 3 spatial dimensions instead of, say, 4 or 5?" And "Why are dimensions separated by right angles instead of, say, 45 degrees or 109.5 degrees?" While string theory has some ideas along this line (that I don't understand), I'm pretty sure the strict answer is "Nobody really knows."

Answer 2

The modulus of a vector is a scalar quantity, and it does not depend on the coordinates.

That doesn't mean you don't use them to calculate vector. It means that you can CHOOSE the coordinates you want, but any coordinates you choose will end up adding up to the same value.

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If you are asking why there are components, that's because superposition theorem means that any force acting on any direction produces the same effect as if we replace that force by its components.

The same applies to any vector. For velocity, for example, it is equivalent to describe a varying velocity or a sum of the movements there are in.

Answer 3

Dude156, I asked me the same question a time ago, and, for me, the only thing that put some order in this mess was to start thinking through a different angle. I started wondering if a math deduction that would lead us to a physical law in fact exists. Then, I realized that fundamental laws of physics do not have mathematical deductions. I needed to understand deeply what math is and what physics is.

SHORT ANSWER:

Based on axioms such as what is a point, a line, a plane etc. mathematicians created Trigonometry (lots of corollaries and formulas). At some time, mathematicians created Arithmetic and Algebra (another bunch of corollaries, and formulas about how to sum, multiply, find the x on an equation etc.). At some point they created the vectors (a math object), and started defining the rules attached to these objects. Then, they used Arithmetic, Algebra and Trigonometry to write a vector as a sum of it components. That is the answer to "Why are we allowed to use trigonometry to get vector components". It is a matter of math, not physics. But if your question is why velocity and forces can be treated as vectors, the answer is because when we trate them like that, we can make useful conclusions, just it.

LONG ANSWER:

I'm not too loyal to books, so what comes next in this post isn't exactly written on them. I just try to find arguments that feels like true for me, and I'm always opened to review them. So, you don't need to agree with what i'll try to say to you.

The math and physics can't EXPLAIN all the phenomena. At the deepest level, they are just representing on a paper a simple transcription of what we are seeing in nature. So, when people say that something is allowed because of some theorem, it is the same to say "we really don't know why we are allowed to use it, but we can verify in nature that the deepest bases of this theorem is true".

There isn't a math deduction which answer why 4+2 is equal to 2+4. The fact that 4+2 and 2+4 are equal shouldn't be take as a fact, it is a definition. What should be thought as a fact is: if next to me there is a group of green objects, and beside this group there is another group of yellow objects, which is near to you, WE (not me) agree that when I take the group near me and then the group which stands next to you, I would have the same resultant group of objects that I would have if I took the group of objects next to you firstly and then the group of objects next to me. So, a+b=b+a is a definition based on how we experiment nature, that is, there is a intersection between how I experiment nature and how you experiment nature in such way that WE agree that we are experiencing the same thing. And that is why math is also considered a language: something that can transmit a message between two people. Think about it. Also, there is no math deduction which answer the why vectors are summed as they are. The sum of two vectors are defined in a way that resultant vector could represent what we want it to represent. Math is a tool to firstly represent nature, and then we use logic to see further. To do that, we create math objects and we give them custom rules in a way that they could REPRESENT the phenomena abstractly on a paper. After that, we can use these objects and rules defined and create a lot of conclusions, in other words, lots of corollaries, and formulas.

Now, about Physics, before talking about vectors, there is an easier case to think, and it is the Kepler's first law, which says "All planets move about the Sun in elliptical orbits, having the Sun as one of the foci.". Kepler didn't gave us a math deduction to reach the law. He firstly saw the nature, measured it, and realized that nature was acting like some math object that someone has already created: an ellipse (something that some older mathematicians, without thinking about planets, described as one of four conic sections, based on axioms such as what is a point, a line, a plane etc.).

It is not 100% correct to say that Newton used vectors. Newton never said the word "vector" in his book Philosophiæ Naturalis Principia Mathematica, often referred simply as Principia. He used algebra and trigonometry and used some rules that is equivalent to what we today call vectors, which was invented some centuries after him by mathematicians dealing with complex numbers.

We are using math on physics just because it is convenient, just because it has the power to condense the nature in some formulas that, using math logic, we can predict what will happen next on nature. But attention! there is lot's of attempts of laws creation that it's predictions do not correspond to what we measure in nature. When it happens, we discard the law. If the laws do good predictions and failure at some point, we still keep them. Using Kepler's laws, we could predict the movement of planets known at his time, but his laws could not explain the Urano orbits when Urano was discovered. After Kepler, Newton's laws could predict the orbit of Urano, but not the orbit o Mercury, but the general theory of relativity of Einstein could predict all orbits, but the accelerated expansion of universe are not predictable by Einstein's equations.

When a law do some bad prediction and still do a lot of good predictions, we keep it because it is useful. Although Newton laws do not predict Mercury's orbit, his laws are useful to predict the behavior of buildings, cars, bridges, and also to send a man to the moon. Now we know that If you're studying the orbit of Mars around the sun without any other very heavy object near them, you could use Kepler's laws.

So, what drive me crazy today and also some people I asked about is why nature act as it was obeying some law. This question, Presocratic philosophers and famous physicists not answered us.