This is one of the most fundamental questions any learner comes across in their journey of understanding the world through mathematical lens (the only reasonable lens there is? What is meant by reason anyway?).
It just occurred to me that the way in which these terms are usually defined seems to have a potential to portray a slightly different view than what they actually are.
Most commonly I’ve seen the terms scalars and vectors defined as follows.
Scalar: A scalar is a quantity that has only magnitude (size).
Vector: A vector is a quantity that has both magnitude (size) and direction.
For instance, looking up “scalar vs vector” on the Internet, the top three websites in the search results define them as below.
These are all great definitions for developing intuitions. However, to develop a mathematically rigorous view, some effort is needed on grasping the exactness of these terms.
The analogies like “magnitude vs. magnitude and direction” have an important role in the process of understanding these concepts and why we need them. For a deeper understanding, in addition to these analogies, it is important to see scalars and vectors as they are — detached from the frame we usually see them from. For instance, consider the following definitions:
Scalar: A number, typically real.
Vector: A set of numbers, typically real.
Vectors are nothing but a bunch of scalars. That’s pretty much it! If we look at these terms as they are, they no longer appear too distinct from each other. One is a number, the other is a bunch of numbers.
It may help to think about some questions like:
- What is a direction anyway?
- Why scalars don’t have a direction?
- Why vectors have a direction? Should vectors have to have a direction?
- In reference to what vectors have the property called direction?
I feel that paying too much attention to meta words like “direction” can slightly distort our view and make these terms look more different than they are from each other. The mathematical view of the world consists of only numbers and some operators. What’s more, the attributes like directions are represented mostly using numbers!
The direction is another way to describe an aspect of the relationship between two bunches of numbers (vectors). It implicitly makes some silent assumptions, for example,
- The ordering of numbers has some meaning, and
- The same frame of reference is assumed to represent the two vectors.
But when we say number vs. a bunch of numbers, we aren’t making any unspoken assumptions. That’s why I think it is less prone to being misinterpreted. Simplifying the view in such a manner is more precise and is as useful as the view that’s generally discussed.
