In physics, vectors are quantities completely specified by magnitude and direction only. But why would physicists and mathematicians be interested in such quantities?
In this article I will show that vectors arise naturally as we try to understand the physical universe. And that the special feature of direction is both fundamental and consequential in developing physical quantities.
It all starts with measurements
Suppose I ask you, “How many buttons does your shirt have?” You might say “eight”. That would be a sufficient, satisfactory answer. Here, you have quantified the number of buttons on your shirt by a bare number 8. And that was enough for me without any additional information.
A “quantity” simply means a (real) number.
Suppose I ask again, “how much do you weigh?” You might say “70”. To which I will look at you confused. “70 what? Kilograms?” Here a bare number alone is not sufficient to give me the information I need. This time a number ought to be supplemented with a unit. Such is an example of a physical quantity.
What is a physical quantity?
Quantities that are regarded as physical quantities in physics are those that can be quantified by measurement and assigned to it a numerical value along with a unit. This requirement eliminates plenty of other quantities that may be of interest to people such as those involving emotions and feelings. [No, you cannot use physics to quantify love!]
What is a measurement?
Today, to measure simply means to assign a number and a unit. Whether it’s your height or weight or age, you expect a number not … a color!
However, numbers and units are not all that there is. For example, if I asked you, “How far is your home?” You might say “2 miles”. To which I would look puzzled and ask “in which direction?” 2 miles is indeed a physical quantity, but in this particular case, it is not sufficient.
So measurement isn’t about assigning a number and unit alone, although that is still a fundamental part of it. Measurement is about obtaining information.
The need for measurement arises because we wish to know certain information about the property of a body or environment that we intend to use down the line. Today people use measurements for all sorts of information. Usually, that information comes in the form of a number.
Sailors, for example, are interested in wind speed data. What they are actually interested in the information that they can extract from the wind speed data for their navigation purposes. And for that data to be of any use to them, it must contain both magnitude (the number that depicts wind speed) and the direction of the wind. When the speed of the wind is accompanied by direction, it is known as the velocity of the wind.
The analysis of physical quantities is abstract in nature. We are only interested in the eventuality that measurement was done and a physical quantity was obtained. As a consequence of this abstraction, only two things are of fundamental importance to physical quantities: magnitude (the numerical value) and direction.
The two main types of physical quantities
Physical quantities are then divided into two broad categories: Those requiring magnitude only and those requiring magnitude and direction only.
Those requiring magnitude only are known as scalar quantities and physical quantities requiring both magnitude and direction are known as vector quantities. In this article, we will briefly discuss both and see how they apply to physics.
What are scalar quantities?
Scalar quantities or simply as “scalars” are quantities completely specified by magnitude alone. Quantities such as mass, temperature, volume, and distance do not need direction to be specified, just plain numbers with appropriate units.
These quantities obey ordinary rules of algebra operation; for example, a 3 kg block and a 5 kg block taken together have a combined mass of 8 kg, regardless of the arrangement.
What are vector quantities?
Unlike scalars, these quantities are completely specified by magnitude as well as direction. These quantities do not obey ordinary algebraic rules of operation; they have their own rules of operations.
The velocity of wind as we have discussed previously is a good example of a vector quantity. Other vector quantities include force, acceleration, linear, angular momentum, torque, electric and magnetic fields.
How can we tell if a quantity is a vector or not?
The short answer is; you can’t tell by just looking at the quantity. This is because, as you will learn in the next article, there is a bit more to vectors than just magnitude and direction. The best you can do is to make a guess on whether a quantity is a vector or not based on the necessity for the direction of the quantity.
In order, to be sure whether a quantity is a vector or not, that has to be inferred from experiments (or by just looking it up!).
How do we present vector quantities?
Vector notation
One of the best attributes of vectors is that it presents a mathematical model in which all vector quantities are set up, manipulated, and combined with each other and scalar quantities. Such a system requires a clear notation to represent vector quantities to distinguish them from scalar quantities.
In most printed texts, vectors are presented using boldface letters, e.g. a, A (the style used here). Another common notation is lettering with an arrow above.
Scalar quantities such as mass can, therefore, be presented simply as m whereas a vector quantity such as force is presented as F.
Drawing of vectors: Geometrical presentation
They say a picture is worth a thousand words. That is absolutely true when it comes to vectors. The geometrical representation of vectors is an excellent way to conceptualize how vectors relate to each other.
There is an “innate” temptation to present physical quantities – especially vector quantities geometrically. even for kids, it is much easier to sketch a car in motion than to describe one. Don’t let the word “geometry” turn you off, in this context it simply means to draw.
Vectors are usually presented in a drawing using arrows. This may first come off as abstract to many but it is remarkably natural and spontaneous. Most of the things that we feel obliged to represent using arrows in a drawing are usually vectors. Whenever there is a need to indicate a direction, arrows come very handily.
This idea has been around for some time. Around 300 BC the early Greeks conceived a method to combine algebra and geometry into a single framework. Their central conception was representing numbers geometrically using line segments.
In this system, the length of the line segment would correspond to the number being represented. For example, a segment representing number 10 was twice as long as that representing number 5. They even coined the term “magnitude” for the length of the line segment – a term still used to describe physical quantities today.
Today, vectors are represented geometrically using arrow-headed line segments, just as the Greeks first conceptualized. The length of the line segment (or magnitude as they called it) is proportional to the number representing the quantity. The arrowhead is used to indicate the direction of the vector quantity, which the line segment represents.
Simply put, vectors are presented geometrically by arrows that are drawn to scale to represent the vector quantity. The arrows are usually named symbolically to represent the name of the vector.
