Is it incorrect to conceive ordered list of $n$ real numbers (element of $\mathbb{R^n}$) as points rather than vectors?

Question

I am currently reading this book by Hubbards Linear algebra, vector calculus unified approach and in the beginning it distinguishes between vectors and point; Vectors are related to change, increment or displacement while Points are positional or related to state of given system.

Now here it says as we know that $\mathbb{R^n}$ can be manifested as a point or a vector depending on notation and type of data. Here in one section it particularly mentions that in linear-algebra, one should think of elements of $\mathbb{R^n}$ as vectors. Also, adding points doesn't make sense but vectors make (analogy of adding two positions with two points and adding both increments with vectors).

If we go by above premise than by rules of vector space Rules of addition and scalar multiplication, how can a point be part of a vector space, since it doesn't make sense to add them?

Then how can one explain formation of a plane (euclidean or Cartesian plane pr space). Does affine space points and vector explain Cartesian space which explicitly describes this separate entity called points along with vectors?

Answer 1

The component of a vector are the coordinates of its terminal point when the initial point is the origin.

That is why a point could be considered as the terminal point of a vector in standard position, therefore the coordinates become the components.

One has to pay attention to the context in order to stay consistent.

Answer 2

The difference between a Euclidean space consisting of points and a vector space consisting of vectors is that a vector space has a particular point that is the origin. The vector space axioms require an additive identity vector, which is the origin. You can do all of Euclidean geometry, which is about points, without having an origin. The line segments of Euclid do not refer to an origin as there is no special point in the plane.

I would answer the question in the title that it is not incorrect to think of $n-$tuples of reals as points but you may be giving up something valuable. You can prove that the $n-$tuples of reals form a vector space over the reals. We know a lot about how vector spaces behave. You may be able to use some of that to solve your problem. If you view the $n-$tuples as points you still have a metric space but you lose the origin. That may or may not be important to the problem at hand. If it is not important, viewing them as points is no problem at all.