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The things that goes on in my head
There is a reason why I almost never respond to the question "What are you thinking about?"
If I am allowed a nerdy moment or two.
When I was in elementary school, I'd often play with absolutely no care about my school bus. As a consequence, I often had to walk from school to my house. At the beginning, I took the easiest path to remember (figure a). Just one turn and I'm home.
However, as I grew a little older, I took the more adventurous path (figure b).
By the time I reached the sixth grade, I began to wonder, which is the more efficient path? I drew figure c and analyzed the problem.
And I figured that the sum of the a1 to a3 segments plus the sum of the b1 to b3 segments is equal to the sum of a(total) and b(total). The easy to remember path and the adventurous path would have me taking roughly the same number of steps.
But wait, what if I took a crazier path (figure d)?
Same analysis, same conclusion. In fact, I figured, even if I subdivided the segments into tinier segments, I would still be taking the same number of steps. But what if I subdivided the path into a1, a2, a3,.... a(n) and let n approach infinity? Wouldn't be my path now be a straight line? And isn't the nearest distance between two points, according to Euclid, in fact, a straight line? It is not possible of course, given the houses that would block my way. But here's the thing, according to the Pythagorean Theorem (figure e), C2=a2+b2, and NOT c=a+b. So what's wrong with my analysis? Is it even wrong?
By the time I reached college, I still wondered. And to further complicate things came a lesson in physics on the summation of vector quantities. Getting the sum of vectors along the same line is a simple matter of addition. However, once the vector quantities are at an angle relative to each other (not necessarily right angles), then triangular equations come into play. The question is, WHY? What happened to the "lost" vector quantity? Why do we lose vector quantities when they are at an angle relative to each other? Where does it go?
There is no punchline to this post. Sorry. But now, at least, you know not to ask me "What are you thinking about?"
I don't wanna be a math geek but your problem was really interesting. It reminded me of zeno's paradox. While your problem is certainly a lot different from his (this is the first time I've seen your problem, btw, in all my years dealing with math), both were lacking the same thing - calculus.
ReplyDeleteSolution to Don's TRIGONOMETRIC problem:
1. A(total) + B(total) is not equal to C(total) according to pythagoras
2. You subdivided A(total) into infinite segments and did the same to B(total).
3. This is called differentiation (OMG, I am such a geek)
4. A differential (dX) is an infinitely small increment in a variable (X)
so now you have d(A) and d(B)
If you travelled from your school to your house in the paths d(A)1 then d(B)1 then d(A)2 then d(B)2 and so on and assuming that you can switch directions in an infinitely fast speed, you would arrive at your house in the same time as it took you when you traversed A(total) and B(Total).
The only instance where you will be able to traverse it in the time it takes you to traverse C(total) is if you slightly changed your direction in the first place and traveled in the direction of d(C) or the differential of C which, of course, is equal to the square root of d(A)^2 + d(B)^2
d(C)^2 = d(A)^2 + d(B)^2
so, for example, if you wanted to walk, say, 2 feet in the direction of A, and 2 feet in the direction of B, and you wanted to do it by walking d(A)1 then d(B)1 and d(A)2 then d(B)2, and the total time it takes you to switch directions is equal to the total time it takes a normal person to switch directions once (I don't know how you'll do it though), then you would still reach your goal in the same time it takes you to walk 2 feet in the direction of A and 2 feet in the direction of B.
The problem originated from the thought that travelling d(A) then d(B) is the same as travelling d(C) - which is false. Travelling d(C) is the same as travelling the square root of the sum of d(A)^2 and d(B)^2. The Pythagorean theorem holds.
Is it still true that the shortest distance between your house and school is a straight line? Yes. However, that straight line is the sum of d(C) and not the sum of d(A) and d(B).
Euclidian Geometry still holds true.
That was a real nice problem though. It's the first time I've seen Geometry pitted against Trigonometry. Kung gusto mong mantrip sa mga pa-scientific effect sa inuman, Ask them if the pythagorean theorem is true and then if the shortest distance between two points is a straight line then BAM! Show your problem. Very few engineers would be able to show you the solution to that. Hahaha.
Sorry for the loooong comment.
Peace out, I'm out.
CLASHMAN
I don't quite understand your question regarding vector analysis. Vector quantities are not lost. FOr example, the straight path from your school to your house is a vector (vector C). A vector, btw, implies both magnitude and direction. The vector from your school to the corner is vector A, and from the corner to your house, vector B. Because of this, vector algebra can simply write this as C=A+B (in bold letters). This is because vector algebra has both magnitude and direction. boldface(C) actually means (C)cis(theta) where cis stands for cosine-imaginary-sine and theta is the angle between vector C and vector A.
ReplyDeletevector A, therefore is actually vector (C)cos(theta)
vector B, now would be vector (C)sin(theta)
Vector analysis is just a shorthand we use. If we are going to use trigonometry instead of vector analysis, electromagnetics equations would probably be half a page long instead of being just two lines. But I'm happy to say that it is consistent with geometry and trignometry, in fact, it was derived from these. :)
CLASHMAN
Oh thanks Clashman for geeking up the blog. Finally, I find someone who appreciates the "humor" in the problem I wrote here. Medyo napataas ata IQ requirement on this particular blog post. A lot of chem students get the "lower meniscus ang tagay" joke. But that's too easy. Lol. I don't know how many people can get the "joke" here. :)
ReplyDeleteahaha. i'm like this myself sometimes. the other day, i was lunching with an all-girl group and spaced out while they were talking about dogs, shopping and celebrity politics. When one of them asked me what I was thinking about, I knew it wasn't the crowd where I can ask, "why don't we just shoot off dirt in the atmosphere where the gravity is negligible compared to the upward force of the air so that we could cut the sun's heat in half?". I pretty much wriggled my way out of the situation by commenting on how hard it is to eat noodles with chopsticks.
ReplyDeleteAnyhow, in case you're interested, I figured out (later on) that the answer was because it is too expensive and the dirt will have to come down eventually due to density issues, so we would have spent a fortune to have a rain of mini-bullets made from dirt. At which point, I said, "Maybe we just need more action stars in the senate". :p
Clashman
Kindred spirit! I can't wait for you to be ready with your own blog. I will confess now that I kinda cheated with this one. I banked a lot of time-neutral material before I posted the first entry. That way, when my schedule gets all clogged up, all I have to do is dip into the banked material to keep the blog fresh.
ReplyDeleteAs to the crazy thoughts that goes on in my head, this is exactly the reason why I blog. I had (a now ex-)girlfriend before who kinda got me. But since we broke up, I no longer have that release. I can no longer text her "So what exactly is the power of the one ring? Why doesn't Frodo just use it? Why did it screw up Smeagol?"
"I don't quite understand your question regarding vector analysis. Vector quantities are not lost..."
ReplyDeleteI deliberately avoided addressing this part of your reply, for fear of geeking up the blog too much! :) But what the hey.
I know that the math as far vector addition is concerned is true. It is true because it reflects what is observed in nature/reality. That is what math is supposed to be. It is a tool used to describe reality (well, practical math anyhow).
However, I still wonder, why is this so? Why is nature that way? I can understand why vector quantities would cancel each other out -- if they are in opposite directions. But how come 1 unit + 1 unit is not necessarily 2 units (and this is only true when the vectors are perfectly aligned and along the same direction)? Why? Why is one apple plus one apple not necessarily equal to two apples? At right angles to each other, why is it 1.41 apples? And we are told that the x component of a vector should not affect the y and z components (and vice versa).
Then my brain drifts back to walking from school to my house. LOL!
Sorry. I didn't see this reply soon enough. Hehe. Thanks for the blogging tip. It makes a lot of sense. Anyhow, regarding your question about vectors...
ReplyDeleteScalar - magnitude only
Examples: temperature, quantity or mass, charge
100 degrees Celsius, 40g, 48 coulombs
Vector - magnitude and direction
Examples:displacement, force
your own example from your house to school, 48 Newtons (towards the earth)
Vectors represent not only magnitude but also direction. So apples at right angles to each other cannot be 1.41 apples. Quantity is a scalar term and as such, must be added up using scalar mathematics. So even at right angles, the answer would be 2 apples. As to why the x component should not affect the y and z components, it depends on what the instructor was talking about. For example, your path from school to house can be resolved into to x an y (let's say the x is horizontal and y is vertical) components (no z components there). If directly above your house, there's a mall, then it is obvious that the path from the school to the mall would have a bigger y component but same x component. In this regard, addition to the y component did not change the x component. It would, however, change the resultant vector. In this case, the path would be changed from "school to house" to "school to mall". The resultant changes with the change of the components but the other components are not affected.
Clashman
Geez, I can't imagine how time has flied for me. I mean, i used to be a total geek back in school. Now, I can't understand half of the things you guys are talking about.
ReplyDeleteMaybe I did change the day I resigned from being smart. Hehehe.
Now, it feels as if its a life time ago.