Why Vectors?
Until now in your math education you have mostly dealt with numbers or variables which are used to represent many things. Those numbers may represent quantities like: Length, area, density, temperature, height, cost, population size, or any number of other quantities that are adequately represented by a number. The general term for a number is a scalar - or a quantity that represents "size" or "scale" of something. Therefore we can say "Length and temperature are scalar quantities".
Many of the quantities that we use to describe the laws of nature in physics, however, are not able to be described by scalars. Rather, quantities like position, velocity, force, momentum, gravitational field, and many others, require more information than that which a scalar can provide. The reason for this is that all those quantities have direction associated with them as well as a magnitude (or scale).
Vectors are mathematical entities with both a magnitude and a direction, and are used for such purposes in physics. For this reason we need to develop a strong command of vectors, the different representations of vectors, and the associated mathematics used to manipulate vector quantities and expressions. That is the goal of this chapter. Mathematics - specifically the mathematics and calculus of vectors - is going to be the language of the physics we do together. Just as you need an adequate command of English vocabulary and grammar to study Shakespeare, so you need an adequate command of calculus and vectors to study physics.
Really?
You may have studied some physics in high school or taken an intro class in college. In such courses you may only briefly (or not at all) have touched on vectors, and it may have seemed that you got along just fine doing physics without vectors even if vectors came up in conversation.
If you are in that category, I will need to repair your understanding of many of the quantities you discussed in that course. I am troubled by the careless treatment of vector quantities in many physics books as if they were scalars. The subsequent misunderstandings of students who are taught that way can be far-reaching.
One example, if you've done physics before, will follow from this question: Is the constant often called the gravitational acceleration a negative quantity, g=-9.8m/s2? If you think so, you are mistaken. This quantity is not an acceleration in principle (but a field strength) and it is also a vector. A vector is never negative. It has a positive magnitude and a direction. The name and sign associated with that downward direction is up to you. It could be along your negative y axis, but you could just as well call it the positive x axis and still get all the physics right in the end of your calculations. These sorts of misconceptions are ones that I intend to fix (if need be) or avoid altogether if you've never had physics before.
We will develop a thorough understanding of the mathematics and the physics by using the language of vectors and calculus from the start so that there is no room for such error, and then you won’t accidentally treat vectors as scalars.
The reason physics books often treat vector quantities like scalars is that they are really focusing on a single component of a vector, and are therefore limiting their discussions to problems that can be seen as being either physically or mathematically linear or one-dimensional. We will treat those sorts of problems, but also many others that are not linear. Either way we will use the same mathematics of vectors.
It is my impression that learning and then using careful notation may seem burdensome initially but is actually easier in the long run. The biggest benefit will come in more complex applications - situations where motion is 2D or 3D, or during the derivation of more complicated behaviors like motion in rotating coordinate systems (like living on a rotating planet as we do). I you have a good understanding of the math, those things will be much easier.