Saturday, June 30, 2007

Vectors

I've heard that you guys have already started vectors (so fast!), so here's something to help you understand the concept better. Disclaimer: Some parts of the following were self-formulated, so they may not be conceptually very sound, but it helped me to look at things this way.

In physics, we are extremely interested in physical quantities. These are measurements taken from an object, such as mass, force, speed, displacement, etc. that have relationships amongst each other that leads to an understanding of reality. To a certain extent, these quantities are extremely difficult to define (try asking yourself what is mass, force, and displacement. You will find that many times you only have an intuitive grasp of the concept that cannot be put into proper words!) or are defined based on one another.

To understand the fundamental difference between scalars and vectors (which may seem rather easy, but you will encounter some really confusing quantities as you trot along in your physics career), we must see how we add these quantities together.

Let's look at a scalar quantity, like mass. If we have a piece of plasticine of mass 3 g, and we mash it together with a piece of plasticine of mass 4 g, what is the final mass of the plasticine? Of course, the answer is 7 g. Even without knowing how the two pieces were mashed together and other sordid details, all we just had to do to obtain the final "sum" of the two masses was to add two numbers together.

What this means is that scalar quantities are added together or subtracted from each other in the usual sense: we just add or subtract the required numbers and voila, you get your answer. No new mathematics required here.

How about for a vector quantity? Let's have a look at displacement, since you should already be very familiar with the concept. If I travel a displacement of 3 m and then subsequently travelled a subsequent displacement of 4 m, what is my total displacement?

Clearly, what you require now are some details, and the details that are required are the directions in which the displacement occurred. It is clearly because the direction of the quantity matters that they become vectors. Depending on the directions of travel, the answer could be anywhere between - 1 m and 7 m. Literally anywhere.

What you will immediately realise is that the adding of vectors require some higher level mathematics, whereas for scalars, we just simply use arithmetic to perform our computation.
So what kind of higher level mathematics is required? Well, not much more than what you have learnt in co-ordinate geometry!

Let's look at the above example: say you travelled 3 m east, and then 4 m south. Let's super-impose an x-axis and a y-axis over your displacement, with one unit on the "graph" representing one metre:
Now, to find the total displacement at the end of the journey, we have to find the magnitude of the blue vector (which can be found by Pythagoras' Theorem) and its direction, angle t (which can be found by application of your knowledge of trigonometry). Now, you can see clearly that the blue vector can be easily represented by 3 units east, and 4 units south, or, since we have drawn the axes on, 3 units right on the x-axis, and 4 units down on the y-axis. We can represent such a vector as follows:

The entry at the top represents the x-axis (positive means right, negative means left) and the entry at the bottom represents the y-axis (positive means up, negative means down). The advantage of this notation is that for any vector, if you wish to calculate its magnitude or direction, you can easily obtain it from the values.

What is even more remarkable is what comes next:

When we add the two parts of the displacement together by adding each individual entry to each other, we can get the sum total of the two vectors.

So here's the overall conclusion:

1. Any vector can be expressed in the form of a matrix (that's what we call that fancy notation), with the top entry being the x-axis, and the bottom entry being the y-axis.

2. In order to add vectors together, all we have to do is just add up each individual entry to get the resultant vector, for example,
What you will realise is that for scalars, we just have to add the numbers together. Vectors however, aren't that much harder: we just have to add the various entries up together. So what you will realise is that vectors are nothing but 2 or more scalars being added together at the same time.
Now, since the world is pretty much in 3 dimensions, how do we extend this to reality? Well, if you can imagine your x and y-axis being drawn on a flat piece of paper sitting on your table, just picture a new z-axis sticking directly out of the paper, pointing at you. Vectors will now have three entries, not two, but the principles underlying vector addition and subtraction remain the same.
Vectors are basically a mathematical tool for exploring quantities that are directional. You can not only add or subtract vectors from each other when required, you may also multiply them! Please go ahead and check out the dot product and the cross product. For people like CoffeeCoke who has been bullied into studying higher level physics, this is a must!! The vector is one of the most important mathematical tools used in physics!

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