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
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."
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