How should I think about Vectors?

Question

Should I think about a vector as a collection of scalars that are scaled by basis vectors (e.g. i-hat, j-hat) and then added together? Or should I think of it as a coordinate that marks the location of the head which points away from the tail at the origin. Or is there another more useful way I have not considered?

Answer 1

It is entirely dependent on the context in which you are working in, more specifically the vector space in which those vectors in live.

What you are drawing attention to are vectors in the euclidean space $\mathbf{R}^n$ where i suppose it is perfectly legitimate to think of them as points and or arrows interchangebly, however if you are confronted with more abstract vector spaces such as the set of all polynomials having degree at most $m$ or the set of all $n$ by $m$ matrices then perhaps the geometeric intutition is not so useful.

A point of interest will arise later in your studies where you will find that at all finite dimensional vector spaces of $\dim n$ are isomorphic to $\mathbf{R}^n$ so working in $\mathbf{R}^n$ is as good as working in any other vector space with the same dimesion i.e it is not un reasonable to consider spatial-vectors and polynomials interchangebly.

Answer 2

Whatever makes them easier to work with for you.

None of those views generalise well to more general vector spaces.

I would probably go with points in the plane/space representing the head of an arrow.

Answer 3

So you are speaking about "spatial vectors" which are a special type of vector. Technically, a vector is just an element of a vector space. The integer $5$ , for instance, is a vector in the vector space of real numbers over the field of real numbers.

The idea here is to have a vector simply be a point $(x,y,z)$. Only instead of thinking about it as a point we instead focus on the line from the origin that points to that point.

That is (x,y,z), also known as component form, and the directed line segment from the origin to the point (x,y,z) can both be thought of as being the "same vector". Which interpretation you take is really functional to both your taste and the specific problem at hand. In 2nd law analysis in physics you will almost always wind up wanting the component form.