I loved high school Physics. I loved high school Physics so much I decided I wanted to major in Physics (but decided against it since I didn’t think I could handle all the math, and now I’m a math major. The irony kills me). I loved the theories (for this was merely conceptual physics, and that much I could handle then) and I loved the beauty of being able to see such profound order in the most ordinary chaos. Perhaps it would seem counter-intuitive to base one’s faith of God upon science, but for me it made sense: Without God to create it, how could such magnificence randomly fall into place?
Better were the parallels I saw between faith and fact: For example, in Judaism the minimum gathering of people to pray is ten, a minyan. And in Physics, the number of equal sources of sound required to exactly double the loudness of sound is–take a guess–exactly ten. The awe I reaped from that one realisation was stupendous, and still stuns me today: How beautiful it is, for ten to gather to pray, for their prayers to be doubled and uplifted to heaven?
So it’s in that vein that I’ve come to love physics.
It was not in that vein, however, that I’ve come to love vectors. Because, quite simply, I do not love vectors. Vectors make no sense to me. Rather, I’m still in the process of making sense of vectors.
A vector is a physical quantity, physical quantities being descriptors of physical entities, physical entities being…well, you get what I mean (I hope). Some examples of physical quantities are weight, length, and time (I know “time” is not “physical” in the typical sense of the word, but it’s still a physical quantity). These quantities are scalars–that is, they have one value: Ten pounds, ten feet, ten minutes. Vectors, however, are different: Vectors have two values–magnitude (what we might call the scalar part) and direction.
In high school physics, vectors made no sense.
In college physics, they must start to make sense…or I’ll suffer the consequences of defeat.
Since Monday, when classes began, I’ve been reading my textbook, looking on Wikipedia, looking on Simple Wikipedia, and trying in any way I know how to understand vectors: Asking other students, asking my teacher, asking as much as I can (and talking to myself along the way).
Yesterday I made some progress. During one of my many treks back and forth across campus, when I was reasonably alone, I said to myself, “Vectors are motions. They have direction and distance–the ‘how much’ part of their magnitude. If I walk straight down this path, the vector is the combined direction I’m going in and how far I’ve gone.” Then I turned down another path. “And if I walk straight down this path, I’m another vector–how far I travel plus the direction I’m traveling in.” Then I realised that I was walking on an inclined plane, so each step was itself a vector of slightly different direction corresponding to the slightly different direction I was moving in space.
At that point, I decided to think of something else.
Later in the day: “The wind is a vector. The elements are vectors.” And I continued on in this way until just shy of saying “Everything is a vector.”
Today, however, it made more sense. In fact, I daresay I might say that I think I understand them now. (I love, of course, how whenever I think I’m certain of something, I unintentionally add dimensions of uncertainty to how I state that: For example, on considering to change my major to math: “I’m considering choosing the possibility of maybe becoming a math major.” Or on vectors: I daresay I might say that I think I understand them now. But this is a tangent, and not of the trigonometric sort, so I must proceed.)
Today it struck me: My pen is a vector.
I use two pens to take notes. One has black ink and is thin and narrow, very shiny and sleek. The other is a bit bigger and writes in four colors of ink (color-coded notes are both eye-catching and easier to read), so it’s thicker and longer and bright lime green with a white stripe. So in no ways are these pens the same. However, I had one set down on my notebook, when I felt the call of nature. So as I got up, I set the other on my textbook–and upon seeing both in symmetry, perfectly parallel, I realised: My pen is a vector.
Both pens have a top and a tip–the top corresponding to the tail of a vector (represented graphically as a straight line with an arrow at one end) and the tip corresponding to the vector’s head (the arrow), indicating its direction. And since each pen has a “how long” measurement, each pen therefore has a magnitude coupled with its corresponding direction.
By two-dimensional standards, my pens became equal vectors, sharing direction and sharing magnitude. Then I realised my one pen really is longer than the other, and that it is a lot thicker, so in actuality, these two pens are not equal at all–but they are both still vectors. They have magnitude (“how long” and “how much,” in this case) and they have direction (where they were pointing). And they weren’t in motion, showing that vectors, although easily represented by moving particles (of any size), are not defined solely by being in motion, but instead by having magnitude and direction.
So I think I get vectors now. I think. And hopefully, you do, too–and if you do, or don’t, why not comment and tell me what you think? And if you already knew about vectors and understood them, then why don’t you comment and tell me if I’ve got it right, or if I’m barking up the wrong tree?
‘Cause you know, “up a tree” is also a vector. I think.