Why do vectors and points use similar notations? Alright a vector is suppose to be magnitude with direction or rather in a simple sense, "numbers with direction".
What really gets to me is that why would a point and a vector share similar notation for describing themselves ?
Lets take a point in a 3D space $(2,3,4)$, and then lets take a vector $[2,3,4]$.
Why are they using almost the same notations for describing two different mathematical concepts ? Also how is the direction being denoted in the this notation ? Is the direction from the origin ? Can't that be applied to the point as well ? 
I am really confused about this. To be clear I have already read this question, but my question is why use similar notations in both cases, and also how the direction information is inferred from the present notation ?
 A: We call them vectors because they belong to vector spaces. This has the 'context baggage' of meaning we can add two of them together and multiply them by a constant (and these operations behave as you would expect from the arithmethic analogues).
However, one key ingredient of vector spaces is a set (the other being the aformentioned operations). As such, elements of a vector space are not just vectors; they are first and foremost points (of the set).

Visualizing them as points or as 'arrows' is really a matter of context and convenience.
A: The mathematical notion of vector is much larger than what is initially presented. To a mathematician, you start with an algebraic system $F$ of numbers where you are allowed to add, subtract, multiply and divide as usual. Usual examples are to take $F$ to be the real numbers or the rational numbers or the complex numbers (but not, for example, the integers because you can't divide: $3/2$ is not an integer). Then a vector space is any algebraic system of vectors where you can add two vectors and scalar multiply by elements of $F$.
Examples:
I will let $F$ be the real numbers in all of these examples.


*

*real numbers where $a + b$ and $a\cdot b$ are just the usual addition and multiplication

*pairs of real numbers where $(a,b) + (c,d) = (a + c, b + d)$ and $k(a,b) = (ka, kb)$

*more generally: $n$-tuples $(a_1,\dots,a_n)$ of real numbers where addition and scalar multiplication are defined by
$$ (a_1,\dots,a_n) + (b_1,\dots,b_n) = (a_1+b_1,\dots,a_n+b_n) \text{ and } k(a_1,\dots,a_n) = (ka_1,\dots,ka_n) $$
we call this $\mathbf{R}^n$ or $n$-dimensional euclidean space.

*an arrow from the point $(0,0,\dots,0)$ to $(a_1,\dots,a_n)$. The addition is done via the parallelogram law and scalar multiplication is done by stretching the arrow. If we denote this arrow by $[a_1,\dots,a_n]$ then we see that
$$ [a_1,\dots,a_n] + [b_1,\dots,b_n] = [a_1+b_1,\dots,a_n+b_n] \text{ and } k[a_1,\dots,a_n] = [ka_1,\dots,ka_n] $$
so really we didn't do anything other than change $(, )$ to $[, ]$ and started calling these arrows rather than points.

*a weirder one: polynomials are vectors because we can add polynomials and multiply polynomials by real numbers. If we look at all polynomials of degree less than $2$ (i.e. constant, linear or quadratic) then they all look like $a + bx + cx^2$ with $a, b, c$ real numbers. We can then associate this to a tuple $(a, b, c)$ and then addition and scalar multiplication of such polynomials is done as we did before with $n$-tuples.
The reason we think of these all as vectors, and use similar notation, is to emphasize the similar algebraic properties (being able to add and scalar multiply). In this way, we move away from thinking about concrete vectors and move onto an abstract notion of vector (this is where the terminology "abstract algebra" and "abstract linear algebra" comes from).
This is useful if you want to study the theory of vectors but if you  need to do computations it is also helpful to work with concrete vectors. In applications, the difference between an arrow and a point is very important. For actually adding vectors together, it doesn't make a difference.
A: In fact, vectors are a far more abstract concept. A vector is an element of the set of a vector space, which again is a set with some operators that have some specific properties. It shows, that the set of n-tuple of real numbers with addition and scalar multiplication has those properties. And, more important, that all finit-dimensional vector spaces are isomorph to this space (you could say, that "they're the same").
You are actually adressing a more physical interpretation problem. You can interprete these n-tuples either as points or as directions (which I assume you mean when you're talking about "vectors"). Actually it isn't a real problem, because you can switch between the interpretations whenever you want. Let me give you an example.
Conside space curves: $c: [a,b] \rightarrow \mathbb{R}^3$ and the corresponding tangential vectors $T: [a,b] \rightarrow \mathbb{R}^3, t \mapsto \frac{c'(t)}{|c'(t)|}$. Both are functions from a real intervall to the 3dim-space. But in the first case, you interpret the vectors as points in the space, and the range is what you would think of a curve. However, the tangential vector is interpreted as the direction of the curve in every point. Usually you "glue" this vector to the corresponding point of the curve, but you can also let them be vectors in 0; then you get only a circle arc.
However, there are situation where you need both, an anchor point and a directions. In those cases you just consider two vectors: The first one describes the anchor point, the second one the direction. If you are interested in this topic, see tangential bundles.
