By: Ryan Dunagin
I know what you’re thinking. Before you gloss this over as some complicated upper division math blog, don’t be afraid! Vectors do apply to physics and engineering quite a bit, even the most general. But some starting students see the word vector and immediately think complicated math. The goal of this blog is to elucidate vectors in a simple enough way, so as to not make them sound so terrifying. In their applications, they are more intuitive than you might think. (okay, I guess they do require some scary trigonometry, but that’s all, I swear)
Let’s say you are out in your orchard collecting apples, as your day job. As you sweep across your trees, whenever you see an apple ready for picking you pick it, and add it to your basket of apples. Whatever apples you pick that day you want to sell, so whenever you see a customer come to your farm, you sell them some apples. (We won’t worry about the price you sold them at for this analogy). But not everyone is an apple person. Logically, anybody who hates apples will automatically love oranges. So let’s say you are also collecting oranges. Being organized, you decide to have a separate basket for collecting oranges throughout the day. You don’t want to mix up your apples and oranges, do you? In the same way, you don’t want to mix up your vector components. This relates to vector addition, because each time you collect you are adding to a certain basket of fruit (apples or oranges), and each time you sell you are subtracting from that. Same goes with adding and subtracting vectors (For the sake of argument, let’s say that if you get an unreasonable customer who demands apples even when you told them a thousand times that you have run out of them, you are forces to write an IOU, and end up with “negative” apples.)
So at the end of the day when you are tallying up your total apples and oranges, how would you express this amount in words? Let’s say you were left over with 21 apples and 15 oranges. You wouldn’t say you collected 36 apploranges, would you? No. You would say you have 21 apples and 15 oranges. Believe it or not, your response was in Vector Notation: [21,15]
Furthermore, here is the formula for vector addition. As you might notice, only values of the same components will be added together (which is similar to apples and oranges). Another good way of visualizing this, which you may have learned in some kind of math course, is the tail-to-head stacking method. ‘What is this strange method?’ you may ask me. Well it’s an approach to adding vectors for the geometrically inclined. As you may notice by looking at this parallelogram made, the order that you add vectors DOES NOT MATTER.
Since not all apples are going to be ripe, you’re going to get a certain yield from the apple trees that you have. Let’s say that value is 5 apples per tree. Of course a certain amount of trees are going to exist within your field that can produce apples. Let’s say we have 20 such trees. The question is: after a day of collecting apples how many apples will you collect? Obviously we will get 5*20 = 100 apples. Similarly, if we have 25 orange trees and a yield of 6 apples per tree, we can argue that at the end you will collect 150 oranges. It may not seem like it, but what you just did is called dot product multiplication (well, almost…). If we took the trees vector [20,25] and the yield vector [5,6], then asked for the total number of fruits by the end of the day, you would do something similar. After attaining 100 apples and 150 oranges, we simply take 100+150 = 250 total fruits. Notice how first we had to multiply by components to get the total yields, then we added those numbers together. In fact, you can look at this representation of the dot product and see that the order of the vectors does not matter either!
Magnitude is just a fancy way to say the length of the vector (commonly denoted with an absolute value symbol). Usually when considering lengths of sticks or arrows, we are thinking of some positive value. In terms of the vectors themselves, the only thing that makes them negative (versus positive) is its direction. And that is completely arbitrary! (In other words you can just pick the positive direction as you please and then just say that the opposite direction of that is negative.) What is the easiest way to find the magnitude of a vector, if you only know its components? Well luckily for us, our old homeboy philosopher from ancient Greece, Pythagoras, already figured that out for us, in case we ever developed Cartesian coordinates and needed to use it for science. If you make a right triangle and know the length of its sides, then you can easily figure out the length of the hypotenuse using Pythagorean theorem. (without having to whip out any of the trig function). And this works out because direction are defined to be at right angles) to each other! To help solidify this point, meditate on this diagram of Cartesian coordinates that Descartes came up with. Notice how they are all at right angles to each other? Pretty fancy, huh?
Now here comes the trigonometry: The most treasured directional axis in physics is the positive x axis (don’t ask me why). And their favorite greek letter to represent the angle from the positive x axis is theta. Using the unit circle, and values for sin(theta) and cos(theta), we can find that the x component of a vector is associated with cosine and the y component with sine. Try telling that one to Pythagoras!